Skip to main content

Full text of "Hydrodynamics"

See other formats


hydrodynamics 



Sir Horace Lamb 



ias long been the chief storehouse of information 
of all workers in hydrodynamics . . ." NATURL 




WF 



UNIVERSITY 
OF FLORIDA 
LIBRARIES 







ENGINEERING AND PHYSICS 



LI BRARY 



Digitized by the Internet Archive 
in 2013 



http://archive.org/details/hydrodynamicsOOIamb 



HYDRODYNAMICS 



BY 

SIR HORACE LAMB, M.A, LL.D., Sc.D., F.R.S. 

HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; LATELY PROFESSOR 
OF MATHEMATICS IN THE VICTORIA UNIVERSITY OF MANCHESTER 



SIXTH EDITION 






NEW YORK 
DOVER PUBLICATIONS 



First Edition 1879 
Second Edition 1895 
Third Edition 1906 
Fourth Edition 1916 
Fifth Edition 1924 
Sixth Edition 1932 

FIRST AMERICAN EDITION 1945 

BY SPECIAL ARRANGEMENT WITH 

CAMBRIDGE UNIVERSITY PRESS 

AND THE MACMILLAN CO. 




Library of Congress Catalog Card Number: 46-1891 



Manufactured in the United States of America 



Dover Publications, Inc. 

180 Varick Street 

New York 14, N. Y. 



PREFACE 

THIS may be regarded as the sixth edition of a Treatise on the Mathematical 
Theory of the Motion of Fluids, published in 1879. Subsequent editions, 
largely remodelled and extended, have appeared under the present title. 

In this issue no change has been made in the general plan and arrangement, 
but the work has again been revised throughout, some important omissions 
have been made good, and much new matter has been introduced. 

The subject has in recent years received considerable developments, in the 
theory of the tides for instance, and in various directions bearing on the 
problems of aeronautics, and it is interesting to note that the "classical" 
Hydrodynamics, often referred to with a shade of depreciation, is here found 
to have a widening field of practical applications. Owing to the elaborate 
nature of some of these researches it has not always been possible to 
fit an adequate account of them into the frame of this book, but attempts 
have occasionally been made to give some indication of the more important 
results, and of the methods employed. 

As in previous editions, pains have been taken to make due acknowledg- 
ment of authorities in the footnotes, but it appears necessary to add that the 
original proofs have often been considerably modified in the text. 

I have again to thank the staff of the University Press for much valued 
assistance during the printing. 

HORACE LAMB 

April 1932 






CONTENTS 

CHAPTER I 

THE EQUATIONS OF MOTION 

ART. PAGE 

I, 2. Fundamental property of a fluid 1 

3. The two plans of investigation ........ 1 

4-9. ' Eulerian ' form of the equations *of motion. Dynamical equations. 

Equation of continuity.* Physical equations. Surface conditions . 2 

10. Equation of energy 8 

10 a. Transfer of momentum 10 

II. Impulsive generation of motion 10 

12. Equations referred to moving axes 12 

13, 14. 'Lagrangian' form of the equations of motion and of the equation of 

continuity ........... 12 

15, 16. Weber's transformation 14 

16 a. Equations in polar co-ordinates 15 

CHAPTER II 

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES 

17. Velocity -potential. Lagrange's theorem 17 

18, 19. Physical and kinematical relations of (f> 18 

20. Integration of the equations when a velocity-potential exists. Pressure- 
equation 19 

21-23. Steady motion. Deduction of the pressure-equation from the principle 

of energy. Limiting velocity 20 

24. Efflux of liquids ; vena contracta 23 

24 a. 25. Efflux of gases 25 

26-29. Examples of rotating fluid ; uniform rotation ; Rankine's ' combined 

vortex ' ; electromagnetic rotation 28 

CHAPTER III 

IRROTATIONAL MOTION 

30. Analysis of the differential motion of a fluid element into strain and 

rotation 31 

31,32. . ' Flow ' and ' circulation.' Stokes' theorem 33 

33. Constancy of circulation in a moving circuit . . . . . 35 

34, 35. Irrotational motion in simply- connected spaces ; single- valued velocity- 

potential 37 



viii Contents 

ART. PAGE 

36-39. Incompressible fluids ; tubes of flow. <\> cannot be a maximum or mini- 

mum. The velocity cannot be a maximum. Mean value of $ over 

a spherical surface 38 

40, 41. Conditions of determinateness of 41 

42-46. Green's theorem ; dynamical interpretation ; formula for kinetic energy. 

Kelvin's theorem of minimum energy 43 

47, 48. Multiply-connected regions ; ' circuits ' and ' barriers ' .... 49 

49-51. Irrotational motion in multiply-connected spaces ; many- valued velocity- 

potential ; cyclic constants ........ 50 

52. Case of incompressible fluids. Conditions of determinateness of <£ . . 53 

53-55. Kelvin's extension of Green's theorem ; dynamical interpretation ; energy 

of an irrotationally moving liquid in a cyclic space .... 54 

56-58. ' Sources ' and ' sinks ' ; double sources. Irrotational motion of a liquid 

in terms of surface-distributions of sources 57 

CHAPTER IV 

MOTION OF A LIQUID IN TWO DIMENSIONS 

59. Lagrange's stream -function .62 

60. 60 a. Relations between stream- and velocity-functions. Two-dimensional 

sources. Electrical analogies ........ 63 

61. Kinetic energy 66 

62. Connection with the theory of the complex variable .... 66 

63. 64. Simple types of motion, cyclic and acyclic. Image of a source in a circular 

barrier. Potential of a row of sources 68 

65, 66. Inverse relations. Confocal curves. Flow from an open channel . . 72 

67. General formulae ; Fourier method 75 

68. Motion of a circular cylinder, without circulation ; stream-lines . . 76 

69. Motion of a cylinder with circulation; 'lift.' Trochoidal path under 

a constant force 78 

70. Note on more general problems. Transformation methods ; Kutta's 

problem 80 

71. Inverse methods. Motion due to the translation of a cylinder; case of 

an elliptic section. Flow past an oblique lamina ; couple due to 

fluid pressure 83 

72. Motion due to a rotating boundary. Rotating prismatic vessels of 

various sections. Rotating elliptic cylinder in infinite fluid ; general 

case with circulation 86 

72 a. Representation of the effect at a distance of a moving cylinder by a 

double source . 90 

72 b. Blasius' expressions for the forces on a fixed cylinder surrounded by an 

irrotationally moving liquid. Applications ; Joukowski's theorem ; 

forces due to a simple source 91 

73. Free stream-lines. Schwarz' method of conformal transformation . . 94 
74-78. Examples. Two-dimensional form of Borda's mouthpiece ; fluid issuing 

from a rectilinear aperture ; coefficient of contraction. Impact of 
a stream on a lamina, direct and oblique; resistance. Bobyleffs 
problem 96 

79. Discontinuous motions . 105 

80. Flow on a curved stratum 108 



Contents ix 

CHAPTEE V 

IRROTATIONAL MOTION OF A LIQUID : PROBLEMS IN 
THREE DIMENSIONS 

ART. PAGE 

81,82. Spherical harmonics. Maxwell's theory of poles 110 

83. Laplace's equation in polar co-ordinates 112 

84,85. Zonal harmonics. Hypergeometric series . 113 

86. Tesseral and sectorial harmonics 116 

87,88. Conjugate property of surface harmonics. Expansions . . . 118 

89. Symbolical solutions of Laplace's equation. Definite integral forms . 119 

90, 91. Hydrodynamical applications. Impulsive pressures over a spherical 

surface. Prescribed normal velocity. Energy of motion generated . 120 
91 a. Examples. Collapse of a bubble. Expansion of a cavity due to internal 

pressure 122 

92, 93. Motion of a sphere in an infinite liquid; inertia ' coefficient. Effect of 

a concentric rigid boundary . . . . . . . .123 

94-96. Stokes' stream-function. Formulae in spherical harmonics. Stream-lines 

of a sphere. Images of a simple and a double source in a fiscal 

sphere. Forces on the sphere .125 

97. Rankine's inverse method 130 

98, 99. Motion of two spheres in a liquid. Kinematical formulae. Inertia 

coefficients 130 

100, 101. Cylindrical harmonics. Solutions of Laplace's equation in terms of 

Bessel's functions. Expansion of an arbitrary function . . .134 
102. Hydrodynamical examples. Flow through a circular aperture. Inertia 

coefficient of a circular disk . . . . . . . 137 

103-106. Ellipsoidal harmonics for an ovary ellipsoid. Translation and rotation 

of an ovary ellipsoid in a liquid 139 

107-109. Harmonics for a planetary ellipsoid. Flow through a circular aperture. 

Stream-lines of a circular disk. Translation and rotation of a 

planetary ellipsoid 142 

110. Motion of a fluid in an ellipsoidal vessel 146 

111. General orthogonal co-ordinates. Transformation of V 2 </> . . . 148 

112. General ellipsoidal co-ordinates ; confocal quadrics 149 

113. Flow through an elliptic aperture . . 150 

114,115. Translation and rotation of an ellipsoid in liquid; inertia coefficients . 152 

116. References to other problems 156 

Appendix: The hydrodynamical equations referred to general ortho- 
gonal co-ordinates 156 

CHAPTER VI 

ON THE MOTION OF SOLIDS THROUGH A LIQUID : 
DYNAMICAL THEORY 

117,118. Kinematical formulae for the case of a single body 160 

119. Theory of the 'impulse ' . 161 

120. Dynamical equations relative to axes fixed in the body . . . .162 

121, 121 a. Kinetic energy ; coefficients of inertia. Representation of the fluid 

motion at a distance by a double source 163 

122, 123. Components of impulse. Reciprocal formulae 166 



Contents 



ART. 


124. 




125. 




126. 




127- 


-129. 


130. 




131. 




132- 


134. 


134 


a. 


135, 


136. 


137, 


138. 


139- 


-141. 


142, 


143. 


144. 





PAGE 

Expressions for the hydrodynamic forces. The three permanent transla- 
tions ; stability 168 

The possible modes of steady motion. Motion due to an impulsive couple 170 

Types of hydrokinetic symmetry 172 

Motion of a solid of revolution. Stability of motion parallel to the axis. 

Influence of rotation. Other types of steady motion . . .174 

Motion of a ' helicoid ' 179 

Inertia coefficients of a fluid contained in a rigid envelope . . . 180 
Case of a perforated solid with cyclic motion through the apertures. 

Steady motion of a ring ; condition for stability . . . .180 
The hydrodynamic forces on a cylinder moving in two dimensions . . 184 
Lagrange's equations of motion in generalized co-ordinates. Hamiltonian 

principle. Adaptation to hydrodynamics .187 

Examples. Motion of a sphere near a rigid boundary. Motion of two 

spheres in the line of centres 190 / 

Modification of Lagrange's equations in the case of cyclic motion ; 

ignoration of co-ordinates. Equations of a gyrostatic system . . 192 
Kineto-statics. Hydrodynamic forces on a solid immersed in a non- 
uniform stream 197 

Note on the intuitive extension of dynamic principles .... 201 

CHAPTER VII 



145. 




146. 




147. 




148, 


149. 


150. 




151. 




152, 


153. 


154, 


155. 


156. 




157. 




158, 


159. 


159 


i. 


160. 




161- 


163. 


164. 




165. 




166. 




166 


a. 


167 





VORTEX MOTION 

' Vortex-lines ' and ' vortex-filaments ' ; kinematical properties . . 202 
Persistence of vortices ; Kelvin's proof. Equations of Cauchy, Stokes, 

and Helmholtz. Motion in a fixed ellipsoidal envelope, with uniform 

vorticity ............ 203 

Conditions of determinateness ......... 207 

Velocity in terms of expansion and vorticity ; electromagnetic analogy. 

Velocities due to an isolated vortex . 208 

Velocity-potential due to a vortex . .211 

Vortex-sheets 212 

Impulse and energy of a vortex-system ....... 214 

Rectilinear vortices. Stream-lines of a vortex-pair. Other examples . 219 
Investigation of the stability of a row of vortices, and of a double row. 

Karman's ' vortex-street ' 224 

Kirchhoff's theorems on systems of parallel vortices .... 229 
Stability of a columnar vortex of finite section ; Kirchhoff's elliptic 

vortex 230 

Motion of a solid in a liquid of uniform vorticity 233 

Vortices in a curved stratum of fluid 236 

Circular vortices ; potential- and stream-function of an isolated circular 

vortex ; stream-lines. Impulse and energy. Velocity of translation 

of a vortex-ring 236 

Mutual influence of vortex-rings. Image of a vortex-ring in a sphere . 242 
General conditions for steady motion of a fluid. Cylindrical and spherical 

vortices 243 

References 246 

Bjerknes' theorems . 247 

Clebsch's transformation of the hydrodynamical equations . . . 248 



ART. 


168. 




169- 


-174. 


175. 




176. 




177- 


-179. 


180- 


-184. 



Contents xi 

CHAPTER VIII 

TIDAL WAVES 

PAGE 

General theory of small oscillations ; normal modes ; forced oscillations . 250 
Free waves in uniform canal; effect of initial conditions; measuring of 

the approximations ; energy 254 

Artifice of steady motion . . . . 261 

Superposition of wave-systems ; reflection 262 

Effect of disturbing forces ; free and forced oscillations in a finite canal . 263 

Canal theory of the tides. Disturbing potentials. Tides in an equatorial 

canal, and in a canal parallel to the equator; semi-diurnal and 

diurnal tides. Canal coincident with a meridian ; change of mean 

level ; fortnightly tide. Equatorial canal of finite length ; lag of the 

tide 267 

Waves in a canal of variable section. Examples of free and forced 

oscillations ; exaggeration of tides in shallow seas and estuaries . 273 
Waves of finite amplitude ; change of type in a progressive wave. Tides 

of the second order 278 

Wave motion in two horizontal dimensions ; general equations. Oscilla- 
tions of a rectangular basin ........ 282 

Oscillations of a circular basin ; Bessel's functions ; contour lines. Elliptic 

basin ; approximation to slowest mode ...... 284 

Case of variable depth. Circular basin 291 

Propagation of disturbances from a centre ; Bessel's function of the second 

kind. Waves due to a local periodic pressure. General formula for 

diverging waves. Examples of a transient local disturbance . . 293 

198-201. Oscillations of a spherical sheet of water ; free and forced waves. Effect 

of the mutual gravitation of the water. Reference to the case of a sea 

bounded by meridians and parallels 301 

Equations of motion of a dynamical system referred to rotating axes . 307 
Small oscillations of a rotating system ; stability 'ordinary' and 'secular.' 
Effect of a small degree of rotation on types and frequencies of 

normal modes 309 

Approximate calculation of frequencies . . . . , . .313 

Forced oscillations 316 

Hydrodynamical examples ; tidal oscillations of a rotating plane sheet of 

water ; waves in a straight canal . . . . . . .317 

Rotating circular basin of uniform depth ; free and forced oscillations . 320 

Circular basin of variable depth 326 

Examples of approximate procedure ....... 328 

Tidal oscillations on a rotating globe. Laplace's kinetic theory . . 330 

Symmetrical oscillations. Tides of long period 333 

Diurnal and semi-diurnal tides. Discussion of Laplace's solution . . 340 

Hough's investigations ; extracts and results 347 

References to further researches 352 

Modifications of the kinetic theory due to the actual configuration of the 

ocean ; question of phase . . 353 

225, 226. Stability of the ocean. Remarks on the general theory of kinetic stability . 35£ 
Appendix : On Tide-generating Forces 358 



185, 


186. 


187, 


188. 


189, 


190. 


191, 


192. 


193. 
194- 


-197. 



202, 


203. 


204- 


-205 a. 


205 b. 


206. 




207, 


208. 


209- 


-211. 


212. 




212 


a. 


213, 


214. 


215- 


-217. 


218- 


-221. 


222, 


223. 


223, 


a. 


224. 





xii Contents 



CHAPTER IX 

SURFACE WAVES 

ART. PAGE 

227. The two-dimensional problem ; surface conditions 363 

228. Standing waves ; lines of motion 364 

229. 230. Progressive waves ; orbits of particles. Wave- velocity ; numerical tables. 

Energy of a simple-harmonic wave-train 366 

231. Oscillations of superposed fluids 370 

232. Instability of the boundary of two currents 373 

233. 234. Artifice of steady motion 375 

235. Waves in a heterogeneous liquid 378 

236, 237. Group- velocity. Transmission of energy . 380 

238-240. The Cauchy-Poisson wave-problem ; waves due to an initial local eleva- 
tion, or to a local impulse 384 

241. Kelvin's approximate formula for the effect of a local disturbance in 

a linear medium. Graphical constructions 395 

242-246. Surface-disturbance of a stream. Case of finite depth. Effect of inequali- 
ties in its bed 398 

247. Waves due to a submerged cylinder 410 

248, 249. General theory of waves due to a travelling disturbance. Wave- 
resistance 413 

250. Waves of finite height ; waves of permanent type. Limiting form . . 417 

251. Gerstner's rotational waves 421 

252. 253. Solitary waves. Oscillatory waves of Korteweg and De Vries . . 423 

254. Helmholtz' dynamical condition for waves of permanent type . . 427 

255, 256. Wave-propagation in two horizontal dimensions. Effect of a local dis- 

turbance. Effect of a travelling pressure-disturbance; wave-patterns 429 
256 a, 256 b. Travelling disturbances of other types. Ship-waves. Wave-resistance. 

Effect of finite depth on the wave-pattern 437 

257-259. Standing waves in limited masses of water. Transverse oscillation in 

canals of triangular, and semi-circular section 440 

260, 261. Longitudinal oscillations ; canal of triangular section ; edge- waves . 445 
262-264. Oscillations of a liquid globe, lines of motion. Ocean of uniform depth 

on a spherical nucleus 450 

265. Capillarity. Surface-condition 455 

266. Capillary waves. Group-velocity 456 

267. 268. Waves under gravity and capillarity. Minimum wave-velocity. Waves 

on the boundary of two currents 458 

269. Waves due to a local disturbance. Effect of a travelling disturbance ; 

waves and ripples 462 

270-272. Surface-disturbance of a stream ; formal investigation. Fish-line problem. 

Wave-patterns 464 

273, 274 Vibrations of a cylindrical column of liquid. Instability of a jet . . 471 

275 Oscillations of a liquid globe, and of a bubble 473 



Contents xiii 



CHAPTER X 

WAVES OF EXPANSION 

ART. PAGE 

276-280. Plane waves ; velocity of sound ; energy of a wave-system . . . 476 
281-284. Plane waves of finite amplitude; methods of Riemann and Earnshaw. 
Condition for permanence of type ; Rankine's investigations. Waves 

of approximate discontinuity 481 

285, 286. Spherical waves. Solution in terms of initial conditions . . . 489 

287, 288. General equation of sound-waves. Equation of energy. Determinateness 

of solutions . 492 

289. Simple-harmonic vibrations. Simple and double sources. Emission of 

energy ............ 496 

290. Helmholtz' adaptation of Green's theorem. Velocity-potential in terms 

of surface-distributions of sources. Kirchhoff's formula . . . 498 

291. Periodic disturbing forces 501 

292. Applications of spherical harmonics. General formulae .... 503 

293. Vibrations of air in a spherical vessel. Vibrations of a spherical stratum 506 

294. Propagation of waves outwards from a spherical surface ; attenuation 

due to lateral motion 508 

295. Influence of the air on the oscillations of a ball-pendulum ; correction for 

inertia ; damping . . . . . . . . . .510 

296-298. Scattering of sound-waves by a spherical obstacle. Impact of waves on 

a movable sphere; case of synchronism . . . . . .511 

299, 300. Diffraction when the wave-length is relatively large : by a flat disk, 

by an aperture in a plane screen, and by an obstacle of any form . 517 

301. Solution of the equation of sound in spherical harmonics. Conditions at 

a wave-front ........... 521 

302. Sound-waves in two dimensions. Effect of a transient source ; comparison 

with the one- and three-dimensional cases ..... 524 

303. 304. Simple-harmonic vibrations ; solutions in Bessel functions. Oscillating 

cylinder. Scattering of waves by a cylindrical obstacle . . . 527 
305. Approximate theory of diffraction of long waves in two dimensions. 

Diffraction by a flat blade, and by an aperture in a thin screen . 531 

306,307. Reflection and transmission of sound-waves by a grating . . . 533 

308. Diffraction by a semi-infinite screen ... .... 538 

309, 310. Waves propagated vertically in the atmosphere; 'isothermal' and 'con- 

vective' hypotheses . . . .541 

. 547 

. 554 

556 

558 



311, 311a, 312. Theory of long atmospheric waves 

313. General equations of vibration of a gas under constant forces. 

314, 315. Oscillations of an atmosphere on a non-rotating globe . 
316. Atmosphere tides on a rotating globe. Possibility of resonance 



xiv Contents 

CHAPTER XI 

VISCOSITY 

ART. PAGE 

317, 318. Theory of dissipative forces. One degree of freedom; free anu forced 

oscillations. Effect of friction on phase 562 

319. Application to tides in equatorial canal ; tidal lag and tidal friction . 565 

320. Equations of dissipative systems in general ; frictional and gyrostatic 

terms. Dissipation function 567 

321. Oscillations of a dissipative system about a configuration of absolute 

equilibrium ' 568 

322. Effect of gyrostatic terms. Example of two degrees of freedom ; dis- 

turbing forces of long period 570 

323-325. Viscosity of fluids ; specification of stress ; formulae of transformation . 571 
326, 327. The stresses as linear functions of rates of strain. Coefficient of viscosity. 

Boundary-conditions ; question of slipping 574 

328. Dynamical equations. The modified Helmholtz equations; diffusion of 

vorticity - . 576 

329. Dissipation of energy by viscosity 579 

330, 330 a. Flow of a liquid between parallel planes. Hele Shaw's experiments. 

Theory of lubrication ; example 581 

331, 332. Flow through a pipe of circular section; Poiseuille's laws; question of 

slipping. Other forms of section 585 

333, 334. Cases of steady rotation. Practical limitations 587 

334 a. Examples of variable motion. Diffusion of a vortex. Effect of surface- 
forces on deep water 590 

335, 336. Slow steady motion ; general solution in spherical harmonics ; formulae 

for the stresses 594 

337. Rectilinear motion of a sphere ; resistance ; terminal velocity ; stream- 

lines. Case of a liquid sphere ; and of a solid sphere, with slipping 597 

338. Method of Stokes ; solutions in terms of the stream -function . . . 602 

339. Steady motion of an ellipsoid 604 

340. 341. Steady motion in a constant field of force 605 

342. Steady motion of a sphere ; Oseen's criticism, and solution . . . 608 

343. 343 a. Steady motion of a cylinder, treated by Oseen's method. References to 

other investigations 614 

344. Dissipation of energy in steady motion; theorems of Helmholtz and 

Korteweg. Rayleigh's extension 617 

345-347. Problems of periodic motion. Laminar motion, diffusion of vorticity. 

Oscillating plane. Periodic tidal force ; feeble influence of viscosity 

in rapid motions 619 

348-351. Effect of viscosity on water-waves. Generation of waves by wind. Calming 

effect of oil on waves .......... 623 

352, 353. Periodic motion with a spherical boundary ; general solution in spherical 

harmonics 632 

354. Applications ; decay of motion in a spherical vessel ; torsional oscillations 

of a hollow sphere containing liquid 637 

355. Effect of viscosity on the oscillations of a liquid globe .... 639 

356. Effect on the rotational oscillations of a sphere, and on the vibrations of 

a pendulum 641 

357. Notes on two-dimensional problems . 644 



Contents xv 

ART. P AGE 

358. Viscosity in gases ; dissipation function 645 

359, 360. Damping of plane waves of sound by viscosity ; combined effect of 

viscosity and thermal conduction 646 

360 a. Waves of permanent type, as affected by viscosity alone . . . . 650 

360 b. Absorption of sound by porous bodies 652 

361. Effect of viscosity on diverging waves 654 

362, 363. Effect on the scattering of waves by a spherical obstacle, fixed or free . 657 

364. Damping of sound-waves in a spherical vessel 661 

365, 366. Turbulent motion. Reynolds' experiments ; critical velocities of water 

in a pipe ; law of resistance. Inferences from theory of dimensions 663 

366 a. Motion between rotating cylinders 667 

366 b. Coefficient of turbulence ; 'eddy' or 'molar' viscosity .... 668 

366 c. Turbulence in the atmosphere ; variation of wind with height . . 669 

367, 368. Theoretical investigations of Rayleigh and Kelvin 670 

369. Statistical method of Reynolds 674 

370. Resistance of fluids. Criticism of the discontinuous solutions of Kirchhoff 

and Rayleigh 678 

370 a. Karman's formula for resistance 680 

370 b. Lift due to circulation 681 

371. Dimensional formulae. Relations between model and full-scale . . 682 
371a, b, c. The boundary layer. Note on the theory of the aerofoil .... 684 
37 Id, e, f, g. Influence of compressibility. Failure of stream-line flow at high speeds 691 

CHAPTER XII 

ROTATING MASSES OF LIQUID 

372. Forms of relative equilibrium. General theorems ..... 697 

373. Formulae relating to attraction of ellipsoids. Potential energy of an 

ellipsoidal mass 700 

374. Maclaurin's ellipsoids. Relations between eccentricity, angular velocity 

and angular momentum ; numerical tables 701 

375. Jacobi's ellipsoids. Linear series of ellipsoidal forms of equilibrium. 

Numerical results 704 

376. Other special forms of relative equilibrium. Rotating annulus . . 707 

377. General problem of relative equilibrium ; Poineard's investigation. Linear 

series of equilibrium forms ; limiting forms and forms of bifurcation. 

Exchange of stabilities 710 

378-380. Application to a rotating system. Secular stability of Maclaurin's and 

Jacobi's ellipsoids. The pear-shaped figure of equilibrium . . 713 

381. Small oscillations of a rotating ellipsoidal mass; Poincar^'s method. 

References 717 

382. Dirichlet's investigations; references. Finite gravitational oscillations 

of a liquid ellipsoid without rotation. Oscillations of a rotating 

ellipsoid of revolution 719 

383. Dedekind's ellipsoid. The irrotational ellipsoid. Rotating elliptic cylinder 721 

384. Free and forced oscillations of a rotating ellipsoidal shell containing 

liquid. Precession 724 

385. Precession of a liquid ellipsoid 728 

List of Authors cited 731 

Index 734 



HYDRODYNAMICS 

CHAPTER I 

THE EQUATIONS OF MOTION 

1. The following investigations proceed on the assumption that the matter 
with which we deal may be treated as practically continuous and homogeneous 
in structure ; i.e. we assume that the properties of the smallest portions into 
which we can conceive it to be divided are the same as those of the substance 
in bulk. 

The fundamental property of a fluid is that it cannot be in equilibrium in 
a state of stress such that the mutual action between two adjacent parts is 
oblique to the common surface. This property is the basis of Hydrostatics, 
and is verified by the complete agreement of the deductions of that science 
with experiment. Very slight observation is enough, however, to convince 
us that oblique stresses may exist in fluids in motion. Let us suppose for 
instance that a vessel in the form of a circular cylinder, containing water 
(or other liquid), is made to rotate about its axis, which is vertical. If the 
angular velocity of the vessel be constant, the fluid is soon found to be rotat- 
ing with the vessel as one solid body. If the vessel be now brought to rest, the 
motion of the fluid continues for some time, but gradually subsides, and at 
length ceases altogether; and it is found that during this process the portions 
of fluid which are further from the axis lag behind those which are nearer, 
and have their motion more rapidly checked. These phenomena point to the 
existence of mutual actions between contiguous elements which are partly 
tangential to the common surface. For if the mutual action were everywhere 
wholly normal, it is obvious that the moment of momentum, about the axis 
of the vessel, of any portion of fluid bounded by a surface of revolution about 
this axis, would be constant. We infer, moreover, that these tangential stresses 
are not called into play so long as the fluid moves as a solid body, but only 
whilst a change of shape of some portion of the mass is going on, and that 
their tendency is to oppose this change of shape. 

2. It is usual, however, in the first instance to neglect the tangential 
stresses altogether. Their effect is in many practical cases small, and, inde- 
pendently of this, it is convenient to divide the not inconsiderable difficulties 
of our subject by investigating first the effects of purely normal stress. The 
further consideration of the laws of tangential stress is accordingly deferred 
till Chapter XI. 



The Equations of Motion 



[chap. I 




If the stress exerted across any small plane area situate at a point P of 
the fluid be wholly normal, its intensity (per 
unit area) is the same for all aspects of the 
plane. The following proof of this theorem 
is given here for purposes of reference. 
Through P draw three straight lines PA, 
PB, PC mutually at right angles, and let 
a plane whose direction-cosines relatively to 
these lines are I, m, n, passing infinitely 
close to P, meet them in A, B, C. Let 
p, Pi, P2, Pz denote the intensities of the 
stresses* across the faces ABC, PBG, PC A, PAB, respectively, of the 
tetrahedron PABC. If A be the area of the first-mentioned face, the areas 
of the others are, in order, IA, mA, raA. Hence if we form the equation of 
motion of the tetrahedron parallel to PA we have p x . lA = pl . A, where we 
have omitted the terms which express the rate of change of momentum, and 
the component of the extraneous forces, because they are ultimately propor- 
tional to the mass of the tetrahedron, and therefore of the third order of 
small linear quantities, whilst the terms retained are of the second. We 
have then, ultimately, p—p\, and similarly p = p 2 = p 3 , which proves the 
theorem. 

3. The equations of motion of a fluid have been obtained in two different 
forms, corresponding to the two ways in which the problem of determining 
the motion of a fluid mass, acted on by given forces and subject to given 
conditions, may be viewed. We may either regard as the object of our 
investigations a knowledge of the velocity, the pressure, and the density, 
at all points of space occupied by the fluid, for all instants; or we may seek 
to determine the history of every particle. The equations obtained on these 
two plans are conveniently designated, as by German mathematicians, the 
'Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, although 
both forms are in reality due to Eulerf. 



The Eulerian Equations. 

4. Let u, v, w be the components, parallel to the co-ordinate axes, of the 
velocity at the point (x, y, z) at the time t. These quantities are then 
functions of the independent variables x, y, z, t. For any particular value of 
t they define the motion at that instant at all points of space occupied by 

* Reckoned positive when pressures, negative when tensions. Most fluids are, however, 
incapable under ordinary conditions of supporting more than an exceedingly slight degree of 
tension, so that^ is nearly always positive. 

f " Principes generaux du mouvement des fluides," Hist, dc VAcad. dc Berlin, 1755. 

" De principiis motus fluidorum," Novi Comm. Acad. Petrop. xiv. 1 (1759). 

Lagrange gave three investigations of the equations of motion; first, incidentally, in 



2-6] Eulerian Equations 3 

the fluid; whilst for particular values of x, y, z they give the history of 
what goes on at a particular place. 

We shall suppose, for the most part, not only that u, v, w are finite and 
continuous functions of x, y, z, but that their space-derivatives of the first 
order (du/dx, dv/dx, dw/dx, &c.) are everywhere finite*; we shall understand 
by the term 'continuous motion,' a motion subject to these restrictions. 
Cases of exception, if they present themselves, will require separate examina- 
tion. In continuous motion, as thus defined, the relative velocity of any two 
neighbouring particles P, P' will always be infinitely small, so that the line 
PP' will always remain of the same order of magnitude. It follows that if 
we imagine a small closed surface to be drawn, surrounding P, and suppose 
it to move with the fluid, it will always enclose the same matter. And any 
surface whatever, which moves with the fluid, completely and permanently 
separates the matter on the two sides of it. 

5. The values of u, v, w for successive values of t give as it were a series 
of pictures of consecutive stages of the motion, in which however there is no 
immediate means of tracing the identity of any one particle. 

To calculate the rate at which any function F (x, y, z, t) varies for a 
moving particle, we may remark that at the time t + 8t the particle which was 
originally in the position (x, ?/. z) is in the position (x + u8t, y + v8t, z + w8t), 
so that the corresponding value of F is 

F(x + u8t, y + v8t,z + iv8t, t + 8t) = F+u8t d -^ + v8t~- + w8t~ + 8t%- . 

17 ox oy oz dt 

If, after Stokes, we introduce the symbol D/Dt to denote a differentiation 
following the motion of the fluid, the new value of F is also expressed by 
F+DF/Dt.8t, whence 

DF dF dF dF dF 

Bt = Tt+ U Tx + V dy + W dz ' (1) 

6. To form the dynamical equations, let p be the pressure, p the density, 
X, T, Z the components of the extraneous forces per unit mass, at the point 
{x, y, z) at the time t. Let us take an element having its centre at (x, y, z), 
and its edges 8x, 8y, 8z parallel to the rectangular co-ordinate axes. The rate 
at which the ^-component of the momentum of this element is increasing is 
p8x8y8z DujDt; and this must be equal to the ^-component of the forces 

connection with the principle of Least Action, in the Miscellanea Taurinensia, ii. (1760) [Oeuvres, 
Paris, 1867-92, i.]; secondly in his "Memoire sur la Theorie du Mouvement des Fluides," Nouv. 
mem. de V Acad, de Berlin, 1781 [Oeuvres, iv.]; and thirdly in the Mecaniquc Analytique. In this 
last exposition he starts with the second form of the equations (Art. 14, below), but translates 
them at once into the ' Eulerian' notation. 

* It is important to bear in mind, with a view to some later developments under the head of 
Vortex Motion, that these derivatives need not be assumed to be continuous. 



4 The Equations of Motion [chap, i 

acting on the element. Of these the extraneous forces give pBxByBzX. The 
pressure on the yz-fave which is nearest the origin will be ultimately 

that on the opposite face 

(p + \dp\dx . 8%) By Bz. 
The difference of these gives a resultant — dp/dx. BxByBz in the direction of 
^-positive. The pressures on the remaining faces are perpendicular to x. 
We have then 

p Bx By Bz yc = pBxByBz X — ^-BxBy Bz. 

Substituting the value of DujDt from (1), and writing down the sym- 
metrical equations, we have 

du du du du _ Y 1 dp 
dt dx dy dz pdx' 



•(2) 



dv dv dv dv _ v 1 dp 

dt dx dy dz pdy' 

dw dw dw dw _ 7 1 dp 
dt dx dy dz p dz 

7. To these dynamical equations we must join, in the first place, a 
certain kinematical relation between u, v, w, p, obtained as follows. 

If Q be the volume of a moving element, we have, on account of the 
constancy of mass, 

Dt 
\Dp 1 DQ . 

- P m + Qwr Q w 

To calculate the value of 1/Q .DQ/Dt, let the element in question be that 
which at time t fills the rectangular space BxByBz having one corner P at 
{%, y, z), and the edges PL, PM, PN (say) parallel to the co-ordinate axes. 
At time t + Bt the same element will form an oblique parallelepiped, and since 
the velocities of the particle L relative to the particle P are du/dx . Bx, 
dv/dx.Bx, dw/dx.Bx, the projections of the edge PL on the co-ordinate axes 
become, after the time Bt, 

(l+pSt)8*, d ^ti.Zx, d ^ St. Sec, 
\ dx ) dx dx 

respectively. To the first order in Bt, the length of this edge is now 

and similarly for the remaining edges. Since the angles of the parallelepiped 

* It is easily seen, by Taylor's theorem, that the mean pressure over any face of the element 
5x by 5z may be taken to be equal to the pressure at the centre of that face. 



6-t] Equation of Continuity 5 

differ infinitely little from right angles, the volume is still given, to the first 
order in Bt, by the product of the three edges, i.e. we have 

1 DQ dii dv dw (G> . 

or QDi = dx + dy + dz~ ( } 

Hence (1) becomes 

_s^®4; + S)=° ^ 

This is called the 'equation of continuity.' 

rvu - du dv dw //1X 

1 he expression a" "*" a — ^~2~' ' ' 

which, as we have seen, measures the rate of dilatation of the fluid at the 
point (x,y,2), is conveniently called the 'expansion' at that point. From a 
more general point of view the expression (4) is called the 'divergence' of the 
vector (u,v,w); it is often denoted briefly by 

div (u, v, w). 
The preceding investigation is substantially that given by Euler*. 
Another, and now more usual, method of obtaining the equation of con- 
tinuity is, instead of following the motion of a fluid element, to fix the 
attention on an element BxByBz of space, and to calculate the change pro- 
duced in the included mass by the flux across the boundary. If the centre 
of the element be at (x, y, z), the amount of matter which per unit time enters 
it across the yz-f&ce nearest the origin is 

and the amount which leaves it by the opposite face is 



f pu + \ — '- — Bx j ByBz. 
BxByBz, 



The two faces together give a gain 

d .pu 
dx 

per unit time. Calculating in the same way the effect of the flux across the 
remaining faces, we have for the total gain of mass, per unit time, in the 
space BxByBz, the formula 

(d .pu 3 . pv d . pw\ j j j. 

Since the quantity of matter in any region can vary only in consequence of 
the flux across the boundary, this must be equal to 

^(p BxByBz), 

* I.e. ante p. 2. 



6 The Equations of Motion [chap, i 

whence we get the equation of continuity in the form 

^ + 9 _£V_^ + ^ = (5) 

dt ox Oy oz v 

8. It remains to put in evidence the physical properties of the fluid, so 
far as these affect the quantities which occur in our equations. 

In an 'incompressible' fluid, or liquid, we have Dp/Dt= 0, in which case 
the equation of continuity takes the simple form 

a-M4:=° • « 

It is not assumed here that the fluid is of uniform density, though this is of 
course by far the most important case. 

If we wish to take account of the slight compressibility of actual liquids, 
we shall have a relation of the form 

p = /e(p-po)lpo, (2) 

or plp = l+p//e, ..(3) 

where k denotes what is called the 'elasticity of volume.' 

In the case of a gas whose temperature is uniform and constant we have 
the ' isothermal ' relation 

PlPo = p/po> (4) 

where p , p are any pair of corresponding values for the temperature in 
question. 

In most cases of motion of gases, however, the temperature is not constant, 
but rises and falls, for each element, as the gas is compressed or rarefied. 
When the changes are so rapid that we can ignore the gain or loss of heat 
by an element due to conduction and radiation, we have the 'adiabatic' 
relation 

PlPo = (plpo) y , (5) 

where po and p are any pair of corresponding values for the element con- 
sidered. The constant 7 is the ratio of the two specific heats of the gas ; for 
atmospheric air, and some other gases, its value is about 1*408. 

9. At the boundaries (if any) of the fluid, the equation of continuity is 
replaced by a special surface-condition. Thus at a fixed boundary, the velocity 
of the fluid perpendicular to the surface must be zero, i.e. if l> m, n be the 
direction-cosines of the normal, 

lu + mv + nw = (1) 

Again at a surface of discontinuity, i.e. a surface at which the values of u, v, w 
change abruptly as we pass from one side to the other, we must have 

l(ux — u 2 )-\-m (v 1 —v 2 )+ n(w 1 — w 2 ) = 0, (2) 

where the suffixes are used to distinguish the values on the two sides. The 
same relation must hold at the common surface of a fluid and a moving solid. 



7-9] Boundary Condition 7 

The general surface-condition, of which these are particular cases, is that 
if F(x, y, z, t) = be the equation of a bounding surface, we must have at 
every point of it 

DF/Dt = (3) 

For the velocity relative to the surface of a particle lying in it must be 
wholly tangential (or zero), otherwise we should have a finite flow of fluid 
across it. It follows that the instantaneous rate of variation of F for a 
surface-particle must be zero. 

A fuller proof, given by Lord Kelvin*, is as follows. To find the rate of motion (v) of 
the surface F(x, y, z, t)=0, normal to itself, we write 

F(x + lv8t, y + mv§t, z+nvdt, t + 8t) = 0, 
where I, m, n are the direction-cosines of the normal at (x, y y z). Hence 

7 dF dF dF\ dF n 

Since (l,m,n)=^, ^, -^)+R, 

i dF 

wehave V= -R W (•*) 

At every point of the surface we must have 

v = lu + mv + nw, 

which leads, on substitution of the above values of I, m, n, %o the equation (3 N . 

The partial differential equation (3) is also satisfied by any surface moving with the 
fluid. This follows at once from the meaning of the operator DjDt. A question arises as 
to whether the converse necessarily holds ; i.e. whether a moving surface whose equation 
F=0 satisfies (3) will always consist of the same particles. Considering anv such surface 
let us fix our attention on a particle P situate on it at time t. The equation 3 expresses 
that the rate at which P is separating from the surface is at this instant zero : and it is 
easily seen that if the motion be continuous (according to the definition of Alt. 4\ the 
normal velocity, relative to the moving surface F, of a particle at an infinitesimal distance 
f from it is of the order f, viz. it is equal to G( where G is finite. Hence the equation of 
motion of the particle P relative to the surface may be written 

DUDt=GC. 

This shews that log £ increases at a finite rate, and since it is negative infinite to beo-in 
with (when £= u )> it remains so throughout, i.e. £ remains zero for the particle P. 
The same result follows from the nature of the solution of 
dF dF dF dF 
ct ox dy dz ' (o) 

considered as a partial differential equation in F\. The subsidiary system of ordinary 
differential equations is 

dt — — — dy _ d z 

u v ~~ w ' (6) 

* (W. Thomson) "Notes on Hydrodynamics," Camb. and Dub. Math. Journ. Feb 1848 
[Mathematical and Physical Papers, Cambridge, 1882... , i. 83.] 
f Lagrange, Oeuvres, iv. 706. 



8 The Equations of Motion [chap, i 

in which #, y, z are regarded as functions of the independent variable t. These are 
evidently the equations to find the paths of the particles, and their integrals may be 
supposed put in the forms 

x=fi(a,b,c,t), y=f 2 (a,b,c,t), z=f 3 (a, b, c, t), (7) 

where the arbitrary constants a, b, c are any three quantities serving to identify a particle ; 
for instance they may be the initial co-ordinates. The general solution of (5) is then found 
by elimination of a, b, c between (7) and 

F=yjr(a,b,c), (8) 

where \jr is an arbitrary function. This shews that a particle once in the surface F=0 
remains in it throughout the motion. 



Equation of Energy. 

10. In most cases which we shall have occasion to consider the extraneous 
forces have a potential; viz. we have 

X,Y,Z=-f,-f,-f (1) 

ox dy oz w 

The physical meaning of fl is that it denotes the potential energy, per unit 
mass, at the point (so, y, z), in respect of forces acting at a distance. It will 
be sufficient for the present to consider the case where the field of extraneous 
force is constant with respect to the time, i.e. 9f2/3£ = 0. If we now multiply 
the equations (2) of Art. 6 by u, v, w, in order, and add, we obtain a result 
which may be written 

If we multiply this by Sx By Sz, and integrate over any region, we find 

5<r + 7)^///(.| + .| + .S^* <D 

where T=ifff p(u 2 + v 2 + w 2 )dxdydz ) V = fffClpdadydz, (3) 

i.e. T and V denote the kinetic energy and the potential energy in relation 
to the field of extraneous force, of the fluid which at the moment occupies 
the region in question. The triple integral on the right-hand side of (2) may 
be transformed by a process which will often recur in our subject. Thus, by 
a partial integration, 

llju^- dxdydz = 1 1 [pu] dydz — III p ~- dxdydz, 

where [pu] is used to indicate that the values of pu at the points where the 
boundary of the region is met by a line parallel to x are to be taken, with 
proper signs. If I, m, n be the direction-cosines of the inwardly directed 
normal to any element BS of this boundary, we have Sy8z = ± IBS, the signs 
alternating at the successive intersections referred to. We thus find that 

jf [pu] dydz = — ffpu I dS, 



9-io] Energy 9 

where the integration extends over the whole bounding surface. Transforming 
the remaining terms in a similar manner, we obtain 

^ t {T + V)=j\ p{lu + m v + nw)dS + \\\p(^ + f y + d Qdxdydz. ...(4) 

In the case of an incompressible fluid this reduces to the form 

^(T+V)=j[(lu + mv + nw)pdS. (5) 

Since lu + mv + nw denotes the velocity of a fluid particle in the direction of 
the normal, the latter integral expresses the rate at which the pressures pBS 
exerted from without on the various elements BS of the boundary are doing 
work. Hence the total increase of energy, kinetic and potential, of any 
portion of the liquid, is equal to the work done by the pressures on its 
surface. 

In particular, if the fluid be bounded on all sides by fixed walls, we have 

lu + mv + nw = 

over the boundary, and therefore 

T+F= const (6) 

A similar interpretation can be given to the more general equation (4), 
provided p be a function of p only. If we write 



E 



-/*<)■ (7) 



then E measures the work done by unit mass of the fluid against external 
pressure, as it passes, under the supposed relation between p and p, from its 
actual volume to some standard volume. For example, if the unit mass were 
enclosed in a cylinder with a sliding piston of area A, then when the piston 
is pushed outwards through a space B%, the work done is pA . Bx, of which 
the factor ABx denotes the increment of volume, i.e. of p~\ In the case of the 
adiabatic relation we find 

e= A_(p_pJ) (8) 

7 - 1 \p pj 

We may call E the intrinsic energy of the fluid, per unit mass. Now, recalling 
the interpretation of the expression 

du/dx -f dv/dy + dwjdz, 

given in Art. 7, we see that the volume- integral in (4) measures the rate 
at which the various elements of the fluid are losing intrinsic energy by 
expansion; it is therefore equal to —DW/Dt, 

where W =/// Epdxdydz (9) 

Hence ~ (T + V+ W) = \\p(lu + mv + nw)dS (10) 



10 The Equations of Motion [chap, i 

The total energy, which is now partly kinetic, partly potential in relation to 
a constant field of force, and partly intrinsic, is therefore increasing at a rate 
equal to that at which work is being done on the boundary by pressure from 
without. 

On the isothermal hypothesis we should have 

E = <?log(p/po), (11) 

where c 2 = p /p . This measures the 'free energy' per unit mass. With this 
definition of E we have an equation of the same form as (10), although the 
meaning is different. 

Transfer of Momentum. 

10 a. If we fix our attention on the fluid which at the instant t occupies 

a certain region, the space which it occupies after a time St will differ from 

the original region by the addition of a surface film of (positive or negative) 

thickness 

(lu + mv -f nw) St, 

where (I, m, n) is the direction of the outward normal to the surface. Hence 

it is easy to see that the rate, at time t, at which the momentum of this 

particular portion of fluid is increasing is equal to the rate of increase of the 

momentum contained in a fixed region having the same boundary, together 

with the flux of momentum outwards across the boundary. 

In symbols, considering momentum parallel to Ox, we have 

Jjj g p dxdydz = jjj p g + u g + v g + w If) dxdydz 

p ^- dxdydz + \\ pu (lu + mv + nw) dS 

'Zip") + d(pv) + dip\ s 

k ox 0y oz / 



pu dxdydz + \\ pu (lu + mv + mv)dS, (1) 



- \\\u . 

0y 

_d L 
~ dt 
by Art. 7 (5). 

In steady motion (Art. 21) the first term on the right hand disappears, 
and the rate of increase of momentum of any portion of fluid is equal to the 
flux of momentum outwards across its boundary. 

Conversely, if we apply the above principle to the fluid contained at any 
instant in a rectangular space SxSySz, we reproduce the equation of motion 
(Art. 6). 

Impulsive Generation of Motion. 

11. If at any instant impulsive forces act bodily on the fluid, or if the 
boundary conditions suddenly change, a sudden alteration in the motion may 
take place. The latter case may arise, for instance, when a solid immersed 
in the fluid is suddenly set in motion. 



10— ll] 



Generation of Motion 



11 



Let p be the density, u, v, w the component velocities immediately before, 
u', v r , vf those immediately after the impulse, X', Y', Z' the components of 
the extraneous impulsive forces per unit mass, m the impulsive pressure, 
at the point (x, y, z). The change of momentum parallel to x of the element 
defined in Art. 6 is then p8x8y8z(u — u); the ^-component of the extraneous 
impulsive forces is pSxSySzX' ', and the resultant impulsive pressure in the 
same direction is — dm/dx. SxSySz. Since an impulse is to be regarded as an 
infinitely great force acting for an infinitely short time (t, say), the effects of 
all finite forces during this interval are to be neglected. 

dm 



Hence, 



phxhyhz{u —u) — pSxSySzX' — ^— SxSySz, 



dx 



or 



Similarly, 



X 



v=Y'- 



w 



w 



= Z'- 



\dm_ 
p dx 

p dy 

ldvr 

p dz 



■(i) 



These equations might also have been deduced from (2) of Art. 6, by 
multiplying the latter by Bt, integrating between the limits and r, putting 



( T Zdt, vr={ T pdt, 

Jo Jo 



u — u = — 



V — — 



w 



w = — 



■(2) 



X'=\ Xdt, F r = Ydt, Z' 
Jo Jo 

and then making r tend to the limit zero. 

In a liquid an instantaneous change of motion can be produced by the 

action of impulsive pressures only, even when no impulsive forces act bodily 

on the mass. In this case we have X', Y', Z' = 0, so thafc 

id™ 

p dx 
19z? 

p d v 

Idja 

p dz 

If we differentiate these equations with respect to x, y, z y respectively, and 
add, and if we further suppose the density to be uniform, we find by Art. 8 (1) 
that 

dy* + dz* ~ ' 

The problem then, in any given case, is to determine a value of m satisfying 
this equation and the proper boundary conditions* ; the instantaneous change 
of motion is then given by (2). 

* It will appear in Chapter in. that the value of w is thus determinate, save as to an additive 
constant. 






12 The Equations of Motion [chap, i 

Equations referred to Moving Axes. 

12. It is sometimes convenient in special problems to employ a system 
of rectangular axes which is itself in motion. The motion of this frame may 
be specified by the component velocities u, v, w of the origin, and the com- 
ponent rotations p, q, r, all referred to the instantaneous positions of the axes. 
If u, v, w be the component velocities of a fluid particle at (x, y, z), the rates 
of change of its co-ordinates relative to the moving frame will be 

Dx Dy Dz 

j^ = u-u + ry-qz, j+=v-v + pz-rx, -^ =w- w + qa?-py. ...(1) 

After a time St the velocities of the particle parallel to the new positions 
of the co-ordinate axes will have become 

, (du du Dx du Dy du Dz\ ~ c - /ON 

To find the component accelerations we must resolve these parallel to the 
original positions of the axes in the manner explained in books on Dynamics. 
In this way we obtain the expressions 



(3) 



du du Dx du Dy du Dz 

dv dv Dx dv Dy dv Dz 

dt- VW+ru+ dxDi + dyWt + diDi 

dw dw Dx dw Du dw Dz 

__q M + p„ + __ + __ + ___ 

These will replace the expressions in the left-hand members of Art. 6 (2)*. 
The general equation of continuity is 

dp d ( Dx\ d ( Dy\ d ( Dz\ A 

reducing in the case of incompressibility to the form 

du dv dw _ „ 

dx dy dz ^ 

as before. 

The Lagrangian Equations. 

13. Let a, b, c be the initial co-ordinates of any particle of fluid, x, y, z 
its co-ordinates at time t. We here consider x, y, z as functions of the 
independent variables a, b, c,t\ their values in terms of these quantities give 
the whole history of every particle of the fluid. The velocities parallel to 

* Greenhill, "On the General Motion of a Liquid Ellipsoid...," Proc. Gamb. Phil. Soc. iv. 
4 (1880). 



i2-u] Lagrangian Equations 13 

the axes of co-ordinates of the particle (a, b, e) at time t are dx/dt, dy/dt, dz/dt, 
and the component accelerations in the same directions are d 2 x/dt 2 , d 2 y/dt 2 , 
d 2 zjdt 2 . Let p be the pressure and p the density in the neighbourhood of 
this particle at time t; X, Y, Z the components of the extraneous forces per 
unit mass acting there. Considering the motion of the mass of fluid which 
at time t occupies the differential element of volume BwBySz, we find, by the 
same reasoning as in Art. 6, 

d 2 x _ Y 1 dp 

di 2 p dx ' 

dhj^ldp 

dt 2 p dy ' 

d 2 z _ „ _ 1 dp 
dt 2 ~ pdz' 

These equations contain differential coefficients with respect to oc, y, z, whereas 
our independent variables are a, b, c, t To eliminate these differential coef- 
ficients, we multiply the above equations by dx/da, dy/da, dzjda, respectively, 
and add; a second time by dx/db, dy/db, dz/db, and add; and again a third time 
by dxjdc, dy/dc, dz/dc, and add. We thus get the three equations 



'c 2 x 

M 2 ' 


-*)£+©- 


->)&♦©- 


J da pea 


'd 2 x 
3t 2 ' 


-*)S+@- 


■*)&♦©- 


■ z )i + "i=»' 


'd 2 x 

jt 2 ' 


-*)«+©- 


-')&-©■ 


-z^+l*-o. 

jde pdc 



These are the 'Lagrangian' forms of the dynamical equations. 

14. To find the form which the equation of continuity assumes in terms 
of our present variables, we consider the element of fluid which originally 
occupied a rectangular parallelepiped having its centre at the point (a, b, c), 
and its edges Sa, 8b, Be parallel to the axes. At the time t the same element 
forms an oblique parallelepiped. The centre now has for its co-ordinates 
x, y, z\ and the projections of the edges on the co-ordinate axes are 
respectively 

dx 5 dy * dz . 
fr- oa, TT'Oa, ^oa; 
oa oa oa 

!»■ >■ >■ 

£* i* s* 



14 



The Equations of Motion 



[chap, i 



The volume of the parallelepiped is therefore 



dx 
da 

dx 
db' 

dx 
do' 



da' 

dy 
db' 

dy 
dc' 



dz_ 
da 

dz_ 
db 

dz_ 

dc 



8a 8b 8c, 



or, as it is often written, 



d (a, b, c) 



■(1) 



(2) 



Hence, since the mass of the element is unchanged, we have 

d(x,y,z) _ 

P d(a~J^)- p0 > 

where p is the initial density at (a, b, c). 

In the case of an incompressible fluid p = p , so that (1) becomes 

d(x,y,z) ^ 1 
d(a,b,c) 

Weber's Transformation. 
15. If as in Art. 10 the forces X, Y, Z have a potential 12, the dynamical 
equations of Art. 13 may be written 

d 2 x dx d 2 y dy d 2 z dz _ 312 1 dp „ . 
dt 2 da dt 2 da dt 2 da~ da pda' 
Let us integrate these equations with respect to t between the limits and t. 
We remark that 






'* d 2 x dx 
o dt 2 da 



dx dx 

dt da 
dxdx . d 

dtda~ Uo ~ <i da~ 



f [*dx d 2 x , 
o Jo dtdadt 

[* (dx\ 
Jo \dt) 



dt, 



where u is the initial value of the ^-component of velocity of the particle 
(a, b, c). Hence if we write 



we find 



*4'[J?+ n -*®" + (IHI)*}]* » 



dx dx 



dy dy dz dz d% . 

dt da dt da dt da ° da 



dx dx dy dy dz dz 
dtdb + Jtdb + dtdb 



v = - 



dx dx dy dy dz dz _ 

didc' { "didc + dtdc~ Wo ~ 



db> 

dc' 



•(2) 



* H. Weber, "Ueber eine Transformation der hydrodynamischen Gleichungen," Crelle, lxviii. 
(1868). It is assumed in (1) that the density p, if not uniform, is a function of p only. 



14-16 a] Polar Co-ordinates 15 

These three equations, together with 

3 I=I*--*{©' + (l)' + (l)'} ; <»> 

and the equation of continuity, are the partial differential equations to be 
satisfied by the five unknown quantities x,y, z,p>x'> P being supposed already 
eliminated by means of one of the relations of Art. 8. 
The initial conditions to be satisfied are 

x — a, y = b, z — c, % = 0. 

16. It is to be remarked that the quantities a, b, c need not be restricted 
to mean the initial co-ordinates of a particle; they may be any three quanti- 
ties which serve to identify a particle, and which vary continuously from one 
particle to another. If we thus generalize the meanings of a, 6, c, the form of 
the dynamical equations of Art. 13 is not altered; to find the form which the 
equation of continuity assumes, let x ,y Q , z now denote the initial co-ordinates 
of the particle to which a, 6, c refer. The initial volume of the parallepiped, 
whose centre is at (x , y , z ) and whose edges correspond to variations Sa, Sb, Be 
of the parameters a, b, c, is 

d {yy°'*°h aSbSc, 

d (a, 6, c) 

sothatwehave p |£f*> = Po 3 -^£o) (1) 

r d (a, b, c) ru 3 (a, 6, c) v ; 

or, for an incompressible fluid, 

d(x t y,z) _ d(x 0y y ,z ) 

d(a,b,c) d(a,b,c) VW 

Equations in Polar Co-ordinates. 

16 a. In the preceding investigations Cartesian co-ordinates have been 
employed, as is usually most consistent in the proof of general theorems. For 
special purposes polar co-ordinates are occasionally useful, and the appropriate 
formulae, on the 'Eulerian' plan, are accordingly given here for reference. 

In plane polars we may use u and v to denote the radial and transversal velocities, 
respectively, at the point (r, 6) at time t. Since the radius vector of a particle is revolving 
at the rate v/r, the ordinary theory of rotating axes gives for the component accelerations: 

Du v Dv v 

M-r- V < Di + r- U ' « 

where, by the method of Art. 5, 

I) d d d 

wrdt +u dr +v ri0 (2) 

The ' expansion ' (A) is found by calculating the rate of flux out of the quasi-rectangular 
element whose sides are Sr and rdd ; thus 

du u dv 
A + _ + (3) 

or r rod v ' 



16 The Equations of Motion [chap, i 

In spherical polars we denote the radial velocity at (r, 6, cf>) by u, the velocity at right 
angles r in the plane of 6 by v, and the velocity at right angles to the plane of 6 by w. A 
triad of lines drawn from the origin parallel to these directors, when taken in this order, 
will, on the usual conventions, form a right-handed system. The changes in the angular 
co-ordinates of a particle in time dt are given by 

rdd = v8t, r sin 08(f> = wdt. 

This involves a rotation of the above system relative to its instantaneous position, with 
components 

cos 68(f), -sin<9Sc£, S<9. 

Hence if p, q, r are the components of the instantaneous angular velocity of the system, 
we have 

p = -cot0, q=--, r=- (4) 

The required accelerations of the particle which is at (/•, 6, (f>) are therefore 



.(5) 



Du Du v 2 -\-w 2 
--n- + q W =-^ — , 

Dv Dv uv w 2 

m - vw+ ru= m + 7 --cote, 

Dw Dio wu vw , , 

_-q % + p,= _- + _ + -coU 

where j- = k\+u ^+v-^ + io-—. — ^- (6) 

Dt dt or rdd rsmddcf* 

The expansion is found by calculating the flux out of the quasi-rectangular space whose 
edges are 8r, rb&, rsin 68cf>, and is 

du n u dv v , . dw /h ,. 

dr r rdd r rsm&dcfi 



CHAPTER II 

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES 

17. In a large and important class of cases the component velocities 
u, v, w can be expressed in terms of a single- valued function <£, as follows: 

d<b deb d<b /1X * 

«,„,«—£ - Ty , -f z (i)* 

Such a function is called a 'velocity-potential,' from its analogy with the 
potential function which occurs in the theories of Attractions, Electro- 
statics, &c. The general theory of the velocity-potential is reserved for the 
next chapter; but we give at once a proof of the following important 
theorem : 

If a velocity potential exist, at any one instant, for any finite portion of 
a perfect fluid in motion under the action of forces which have a potential, 
then, provided the density of the fluid be either constant or a function of the 
pressure only, a velocity-potential exists for the same portion of the fluid at 
all instants before or after f. 

In the equations of Art. 15, let the instant at which the velocity- 
potential (f>Q exists be taken as the origin of time; we have then 

u da + v db + w dc = — d<f> , 
throughout the portion of the mass in question. Multiplying the equations (2) 
of Art. 15 in order by da, db, dc, and adding, we get 

^idx+ ^dy+~-dz — (u da + v db -\-w dc)——dx, 

or, in the 'Eulerian' notation, 

udx + vdy + wdz = — d ($ + %) = — d<p, say. 
Since the upper limit of t in Art. 15 (1) may be positive or negative, this 
proves the theorem. 

It is to be particularly noticed that this continued existence of a velocity- 
potential is predicated, not of regions of space, but of portions of matter. 

* The reasons for the introduction of the minus sign are stated in the Preface. The theory of 
' cyclic ' velocity- potentials is discussed later. 

t Lagrange, "Memoire sur la Theorie du Mouvement des Fluides," Nouv. mem. de VAcad. de 
Berlin, 1781 [Oeuvres, iv. 714]. The argument is reproduced in the Mecanique Analytique. 

Lagrange's statement and proof were alike imperfect ; the first rigorous demonstration is due 
to Cauchy, "Memoire sur la Theorie des Ondes," Mem. de VAcad. roy. des Sciences, i. (1827) 
[Oeuvres Completes, Paris, 1882... , l re Serie, i. 38]; the date of the memoir is 1815. Another 
proof is given by Stokes, Camb. Trans, viii. (1845) (see also Math, and Phys. Papers, Cambridge, 
1880... , i. 106, 158, and ii. 36), together with an excellent historical and critical account of the 
whole matter. 



18 Integration of the Equations in Special Cases [chap, ii 

A portion of matter for which a velocity-potential exists moves about and 
carries this property with it, but the part of space which it originally occupied 
may, in the course of time, come to be occupied by matter which did not 
originally possess the property, and which therefore cannot have acquired it. 
The class of cases in which a single- valued velocity-potential exists includes 
all those where the motion has originated from rest under the action of forces 
of the kind here supposed ; for then we have, initially, 

u da + v db + w dc = 0, 

or cj) = const. 

The restrictions under which the above theorem has been proved must 
be carefully remembered. It is assumed not only that the extraneous forces 
X, Y, Z, estimated at per unit mass, have a potential, but that the density p 
is either uniform or a function of p only. The latter condition is violated, 
for example, in the case of the convection currents generated by the unequal 
application of heat to a fluid; and again, in the wave-motion of a hetero- 
geneous but incompressible fluid arranged originally in horizontal layers 
of equal density. Another case of exception is that of 'electro-magnetic 
rotations'; see Art. 29. 

18. A comparison of the formulae (1) with the equations (2) of Art. 11 
leads to a simple physical interpretation of (j>. 

Any actual state of motion of a liquid, for which a (single-valued) 
velocity-potential exists, could be produced instantaneously from rest by the 
application of a properly chosen system of impulsive pressures. This is evident 
from the equations cited, which shew, moreover, that (f> — vr/p + const. ; so 
that ot = p</> + C gives the requisite system. In the same way ts = — p$ + C 
gives the system of impulsive pressures which would completely stop the 
motion*. The occurrence of an arbitrary constant in these expressions merely 
shews that a pressure uniform throughout a liquid mass produces no effect 
on the motion. 

In the case of a gas, (f> may be interpreted as the potential of the extraneous 
impulsive forces by which the actual motion at any instant could be produced 
instantaneously from rest. 

A state of motion for which a velocity-potential does not exist cannot be 
generated or destroyed by the action of impulsive pressures, or of extraneous 
impulsive forces having a potential. 

19. The existence of a velocity-potential indicates, besides, certain kine- 
matical properties of the motion. 

A 'line of motion' is denned to be a line drawn from point to point, so 

* This interpretation was given by Canchy, loc, cit., and by Poisson, Mem. de VAcad. roy. des 
Science*, i. (1816). 



17-20] Velocity-Potential 19 

that its direction is everywhere that of the motion of the fluid. The diffe- 
rential equations of the system of such lines are 

dx dy dz ,_. 

— = — = — (2) 

U V w 

The relations (1) shew that when a velocity-potential exists the lines of 
motion are everywhere perpendicular to a system of surfaces, viz. the 'equi- 
potential 5 surfaces <£ = const. 

Again, if from the point (x, y, z) we draw a linear element 8s in the 
direction (I, m, n), the velocity resolved in this direction is lu + mv + nw, or 

dcj> dx dcj>dy d(j>dz , . , _ d<j> 
dx ds dy ds dz ds' ds' 

The velocity in any direction is therefore equal to the rate of decrease of 
cf) in that direction. 

Taking 8s in the direction of the normal to the surface $ = const., we see 
that if a series of such surfaces be drawn corresponding to equidistant values 
of 0, the common difference being infinitely small, the velocity at any point 
will be inversely proportional to the distance between two consecutive surfaces 
in the neighbourhood of the point. 

Hence, if any equipotential surface intersect itself, the velocity is zero 
at the intersection. The intersection of two distinct equipotential surfaces 
would imply an infinite velocity. 

20. Under the circumstances stated in Art. 17, the equations of motion 
are at once integrable throughout that portion of the fluid mass for which a 
velocity-potential exists provided p is either constant, or a definite function 
of p. For in virtue of the relations 

dv/dz = dw/dy, dw/dx = du/dz, du/dy = dv/dx, 

which are implied in (1), the equations of Art. 6 may be written 

d 2 cj> du dv dw 311 19p 

dxdt ox dx dx dx pox 



These have the integral 



l + W + E = f t +F(t) (4) 



Here q denotes the resultant velocity (u 2 + v 2 -f w 2 )%, F(t) is an arbitrary 
function of t, and E is defined by Art. 10 (7), and has (in the case of a gas) the 
interpretation there given. 

Our equations take a specially simple form in the case of an incompressible 
fluid; viz. we then have 

MS-a-w+m :■■«> 



20 Integration of the Equations in Special Cases [chap, ii 

with the equation of continuity 

w + df + M- 0> (6) 

which is the equivalent of Art. 1 (8). When, as in many cases which we 
shall have to consider, the boundary conditions are purely kinematical, the 
process of solution consists in finding a function which shall satisfy (5) and 
the prescribed surface-conditions. The pressure p is then given by (4), and 
is thus far indeterminate to the extent of an additive function of t. It 
becomes determinate when the value of p at some point of the fluid is given 
for all values of t. Since the term F{t) is without influence on resultant 
pressures it is frequently omitted. 

Suppose, for example, that we have a solid or solids moving through a liquid com- 
pletely enclosed by fixed boundaries, and that it is possible {e.g. by means of a piston) to 
apply an arbitrary pressure at some point of the boundary. Whatever variations are made 
in the magnitude of the force applied to the piston, the motion of the fluid and of the 
solids will be absolutely unaffected, the pressure at all points instantaneously rising or 
falling by equal amounts. Physically, the origin of the paradox (such as it is) is that the 
fluid is treated as absolutely incompressible. In actual liquids changes of pressure are 
propagated with very great, but not infinite, velocity. 

If the co-ordinate axes are in motion, the formula for the pressure is 

-»('*-3H('B-a-'('&-'i9 <" 

where q 2 = (u - u) 2 + (v — v) 2 4- (w — w) 2 (8) 

This easily follows from the formulae for the accelerations given in Art. 12 (3). 

Steady Motion. 

21. When at every point the velocity is constant in magnitude and 
direction, i.e. when 

^ = ^ = ^ = (1) 

dt ' dt v ' dt ' w 

everywhere, the motion is said to be 'steady/ 

In steady motion the lines of motion coincide with the paths of the 
particles. For if P, Q be two consecutive points on a line of motion, a 
particle which is at any instant at P is moving in the direction of the 
tangent at P, and will, therefore, after an infinitely short time arrive at Q. 
The motion being steady, the lines of motion remain the same. Hence the 
direction of motion at Q is along the tangent to the same line of motion, 
i.e. the particle continues to describe the line, which is now appropriately 
called a 'stream-line.' 



20-22] Steady Motion 21 

The stream-lines drawn through an infinitesimal contour define a tube, 
which may be called a 'stream-tube.' 

In steady motion the equations (3) of Art. 20 give 



/ 



^ = _n-ig 2 + constant (2) 



The law of variation of pressure along a stream-line can however in this case 
be found without assuming the existence of a velocity-potential. For if 8s 
denote an element of a stream-line, the acceleration in the direction of motion 
is qdq/ds, and we have 

dq_ dn ldp , 

q ds--fo-~pds-> W 

whence integrating along the stream-line, 

p = -il-W + C. (4) 

This is similar in form to (2), but is more general in that it does not assume 
the existence of a velocity-potential. It must however be carefully noticed 
that the 'constant' of equation (2) and the 'C of equation (4) have different 
meanings, the former being an absolute constant, while the latter is constant 
along any particular stream-line, but may vary as we pass from one stream- 
line to another. 

22. The theorem (4) stands in close relation to the principle of energy. If 
this be assumed independently, the formula may be deduced as follows*. Taking 
first the particular case of a liquid, consider the filament of fluid which at 
a given instant occupies a length AB of a stream-tube, the direction of motion 
being from A to B. Let p be the pressure, q the velocity, fl the potential of 
the extraneous forces, a- the area of the cross-section, at A> and let the values 
of the same quantities at B be distinguished by accents. After a short 
interval of time the filament will occupy a length A-iBx, let m be the mass 
included between the cross-sections at A and A lf or B and B±. Since the 
motion is steady, the gain of energy by the filament will be 

Again, the net work done on it is pm/p — p'm/p. Equating the increment of 
energy to the work done, we have 

p p 

or, using in the same sense as before, 

£ Cl-tf+C, (5) 

P 

which is what the equation (4) becomes when p is constant. 

* This is really a reversion to the methods of Daniel Bernoulli, Hydrodynamica, Argentorati, 
1738. 



22 Integration of the Equations in Special Cases [chap, ii 

To prove the corresponding formula for compressible fluids, we remark that 
the fluid crossing any section has now, in addition to its energies of motion and 
position, the energy ('intrinsic' or 'free' as the case may be) 

-K)--M? : 

per unit mass. The addition of these terms in (5) gives the equation (4). 
In the case of a gas subject to the adiabatic law 

PlPo = (p/po) y , (6) 

the equation (4) takes the form 

i _£_£__, Wg . + a (7) 

23. The preceding equations shew that, in steady motion, and for points 
along any one stream-line*, the pressure is, cceteris paribus, greatest where 
the velocity is least, and vice versa. This statement becomes evident when we 
reflect that a particle passing from a place of higher to one of lower pressure 
must have its motion accelerated, and vice versdf. 

It follows that in any case to which the equations of the last Article 
apply there is a limit which the velocity cannot exceed J. For instance, let 
us suppose that we have a liquid flowing from a reservoir where the velocity 
may be neglected, and the pressure is p , and that we may neglect extraneous 
forces. We have then, in (5), G = po/p, and therefore 

P=Po~ipq 2 (8) 

Now although it is found that a liquid from which all traces of air or other 
dissolved gas have been eliminated can sustain a negative pressure, or tension, 
of considerable magnitude §, this is not the case with fluids such as we find 
them under ordinary conditions. Practically, then, the equation (8) shews 
that q cannot exceed (2p /p)%. This limiting velocity is that with which the 
fluid would escape from the reservoir into a vacuum. In the case of water at 
atmospheric pressure it is the velocity * due to ' the height of the water- 
barometer, or about 45 feet per second. 

If in any case of fluid motion of which we have succeeded in obtaining 
the analytical expression, we suppose the motion to be gradually accelerated 
until the velocity at some point reaches the limit here indicated, a cavity will 
be formed there, and the conditions of the problem are more or less changed. 

It will be shewn, in the next chapter (Art. 44), that in irrotational motion 
of a liquid, whether 'steady' or not, the place of least pressure is always at 

* It will be shewn later that this restriction is unnecessary when a velocity-potential exists. 

t Some interesting practical illustrations of this principle are given by Froude, Nature, xiii. 
1875. 

X Cf. Helmholtz, "Ueber discontinuirliche Flussigkeitsbewegungen, " Berl, Monatsber. April 
1868; Phil. Mag. Nov. 1868 [Wissenschaftliche Abhandlungen, Leipzig, 1882-3, i. 146]. 
0. Reynolds, Manch. Mem. vi. (1877) [Scientific Papers, Cambridge, 1900... , i. 231]. 



22-24] Steady Motion 23 

some point of the boundary, provided the extraneous forces have a potential 
XI satisfying the equation 

d 2 a dHi d 2 n_ 

dx 1 dy 2 dz 2 
This includes, of course, the case of gravity. 

In the general case of a fluid in which p is a given function of p we have, 
putting H = 0, q = 0, in (4), 



-if* W 

Jv 9 



P P 



For a gas subject to the adiabatic law, this gives 

y-1 

v2 _ 2 7 Po 



iS 1 -©' w 



flr- 

7 - 1 p ( \p / 

= ^l(Co 2 -c 2 ), (11) 

if c, = (yp/p)%, = (dp/dp)i, denote the velocity of sound in the gas when at 
pressure p and density p, and c the corresponding velocity for gas under the 
conditions which obtain in the reservoir. (See Chapter x.) Hence the limiting 
velocity is 

2 

Co, 

or 2-214c , if 7 = 1*408. 



t^y 



24. We conclude this chapter with a few simple applications of the 
equations. 

Flow of Liquids, 

Let us take in the first instance the problem of the efflux of a liquid from 
a small orifice in the walls of a vessel which is kept filled up to a constant 
level, so that that motion may be regarded as steady. 

The origin being taken in the upper surface, let the axis of z be vertical, 
and its positive direction downwards, so that Q, — — gz. If we suppose the 
area of the upper surface large compared with that of the orifice, the velocity 
at the former may be neglected. Hence, determining the value of C in 
Art. 21 (4) so that p = P (the atmospheric pressure) when z = 0, we have* 

Z-Z+gg-W (1) 

r r 

At the surface of the issuing jet we have p = P, and therefore 

<? 2 = 2<7*, ■ (2) 

i.e. the velocity is that due to the depth below the upper surface. This is 
known as Torricelli s Theorem^. 

* This result is due to D. Bernoulli, I.e. ante p. 21. 

f "De motu gravium naturaliter aceeierato," Firenze, 1643. 



24 Integration of the Equations in Special Cases [chap, ii 

We cannot however at once apply this result to calculate the rate of efflux 
of the fluid, for two reasons. In the first place, the issuing fluid must be 
regarded as made up of a great number of elementary streams converging 
from all sides towards the orifice. Its motion is not, therefore, throughout 
the area of the orifice, everywhere perpendicular to this area, but becomes 
more and more oblique as we pass from the centre to the sides. Again, the 
converging motion of the elementary streams must make the pressure at the 
orifice somewhat greater in the interior of the jet than at the surface, where 
it is equal to the atmospheric pressure. The velocity, therefore, in the interior 
of the jet will be somewhat less than that given by (2). 

Experiment shews however that the converging motion above spoken of 
ceases at a short distance beyond the orifice, and that (in the case of a circular 
orifice) the jel, then becomes approximately cylindrical. The ratio of the area 
of the section S r of the jet at this point (called the 'vena contracta') to the 
area $ of the orifice is called the 'coefficient of contraction.' If the orifice be 
simply a hole in a thin wall, this coefficient is found experimentally to be 
about '62. 

The paths of the particles at the vena contracta being nearly straight, 
there is little or no variation of pressure as we pass from the axis to the outer 
surface of the jet. We may therefore assume the velocity there to be uniform 
throughout the section, and to have the value given by (2), where z now 
denotes the depth of the vena contracta below the surface of the liquid in the 
vessel. The rate of efflux is therefore 

Vg^.pS' (3) 

The calculation of the form of the issuing jet presents difficulties which 
have only been overcome in a few ideal cases of motion in two dimensions. 
(See Chapter iv.) It may however be shewn that the coefficient of con- 
traction must, in general, lie bet wen h and 1. To put the argument in its 
simplest form, let us first take the case of liquid issuing from a vessel the 
pressure in which, at a distance from the orifice, exceeds that in the external 
space by the amount P, gravity being neglected. When the orifice is closed 
by a plate, the resultant pressure of the fluid on the containing vessel is of 
course nil. If when the plate is removed we assume (for the moment) that 
the pressure on the walls remains sensibly equal to P, there will be an un- 
balanced pressure PS acting on the vessel in the direction opposite to that of 
the jet, and tending to make it recoil. The equal and contrary reaction on 
the fluid produces in unit time the velocity q in the mass pqS' flowing through 
the 'Vena contracta,' whence 

PS = pq 2 8' (4) 

The principle of energy gives, as in Art. 22, 

p=w (5) 



24-24 a] Vena Contracta 25 

so that, comparing, we have S' — \S. The formula (1) shews that the pressure 
on the walls, especially in the neighbourhood of the orifice, will in reality fall 
somewhat below the static pressure P, so that the left-hand side of (4) is an 
under-estimate. The ratio S'/S will therefore in general be > J. 

In one particular case, viz. where a short cylindrical tube, projecting 
inwards, is attached to the orifice, the assumption above made is sufficiently 
exact, and the consequent value ^ for the coefficient then agrees with 
experiment. 

The reasoning is easily modified so as to take account of gravity (or other 
conservative forces). We have only to substitute for P the excess of the static 
pressure at the level of the orifice over the pressure outside. The difference 
of level between the orifice and the ' vena contracta' is here neglected*. 

Another important application of Bernoulli's theorem is to the measure- 
ment of the velocity of a stream by means of a 'Pitot tube.' This consists of 
a fine tube open at one end, which points up-stream, and connected at the 
other end with a manometer. Along the stream-line which is in a line with 
the axis of the tube the velocity falls rapidly from q to 0, so that the manometer 
indicates the value of the 'total head' p + ^pq 2 in the neighbourhood. A second 
manometer connected with a tube closed at the end, but with minute perfora- 
tions in the wall, past which the stream glides, determines the value of the 
'static pressure' p. The density p being known, a comparison of the readings 
gives the value of q. The two contrivances are often combined in one instru- 
ment. The method is extensively used in Aerodynamics, the compressibility 
of the air being found to have little effect up to speeds of the order of 200 ft. 
per sec. 

Floiv of a Gas. 

2A&. The steady now of a gas subject to the adiabatic law presents some 
features of interest. 

Let a be the cross-section at any point of a stream -tube, and hs an element 
of the length in the direction of flow. Omitting extraneous forces we have in 
place of Art. 23 (10) 

*-*- M HSl <■» 

* The above theory is due to Borda (Mem. de VAcad. des Sciences, 1766), who also made 
experiments with the special form of mouth-piece referred to, and found S/S' = 1-942. It was 
re-discovered by Hanlon, Proc. Lond. Math. Soc. iii. 4 (1869) ; the question is further elucidated 
in a note appended to this paper by Maxwell. See also Froude and J. Thomson, Proc. Glasgotv 
Phil. Soc. x. (1876). It has been remarked by several writers that in the case of a diverging 
conical mouth-piece projecting inwards the section at the vena contracta may be less than half 
the area of the internal orifice. 



26 Integration of the Equations in Special Cases [chap, ii 

where the zero suffix relates to some fixed section of the tube. If c be the 

velocity of sound corresponding to the local values of p and p this may be 

written 

? 2 + 2 2 

3 7— 1 X 7— 1 

Again, since the mass crossing any section in unit time is the same, 

pqa- = p q (To (3) 

.p. 1 da _ 1 dq 1 dp dp 

jnence 7 — ~~ 7 ~n ~j 

a- as qds pap as 



--£(-$) « 



It follows from (2) and (4) that in a converging tube q will increase and c 
diminish, or vice versa, according as q is less or greater than c. For a diverging 
tube the statements must be reversed. Briefly, we may say that in a converging 
tube the stream velocity and the local velocity of sound continually approach 
one another, whilst in a diverging tube they separate more and more. 

These results follow also from a graphical representation of the equations (2) and (3). 
Since c 2 is proportional to p? ~ 1 , the latter may be written 

2 2 
cy- 1 qo- = c y- 1 q o- (5) 

If we take abscissae proportional to c and ordinates to q, the equation (2) represents 
an ellipse of invariable shape, drawn through the point (c , q ). For any assigned value of 
ct/o-q the equation (5) represents a sort of hyperbolic curve. For a certain value (</) of o- 
this will touch the ellipse, and we then have q = c. 

The curves A A', BB\ CO' in the annexed diagram correspond to the ratios 



cr 



cr 



7 = 8, 4, 2, 



respectively, whilst the point D corresponds to the 
minimum section a. For still smaller values of o- the 
intersections with the ellipse are imaginary, and steady 
adiabatic flow becomes impossible. The diagram shews 
that for any section greater than a there are two possible 
pairs of values of q and c, as has been remarked by 
Osborne Reynolds and others. 

When q is less than c the representative point on 
the ellipse lies below OB. In a converging tube it 
assumes a sequence of positions such as A\ B', C\ the 
stream-velocity increasing, and the velocity of sound 
decreasing, as the critical section a' is approached. 
When q is greater than c, on the other hand, the repre- 
sentative point lies above OD. In a converging tube 
we have a sequence such as A, B, C ; the stream-velocity 
decreases, and the velocity of sound increases. 




24 a-25] Flow of Gases 27 

25. We consider more particularly the efflux of a gas, supposed to flow 
through a small orifice from a vessel in which the pressure is p and density 
po into a space where the pressure is p x . 

If the ratio po/Pi of the pressure inside and outside the vessel do not exceed a certain 
limit, to be indicated presently, the flow will take place in much the same manner as in 
the case of a liquid, and the rate of discharge may be found by putting p=pi in Art. 23 (10), 
and multiplying the resulting value of q by the area o-i of the vena contracta. This gives 
for the rate of discharge of mass* 



y+i 



»-&M®'-®7~ 



.(6) 



It is plain however that there must be a limit to the applicability of this result ; for 
otherwise we should be led to the paradoxical conclusion that when pi=0, i.e. the discharge 
is into a vacuum, the flux of matter is nil. The elucidation of this point is due to Prof 
Osborne Reynolds f. It appears that qp is a maximum, i.e. the section of an elementary 
stream is a minimum, when a.s appears from (4) the velocity of the stream is equal to tht 
velocity of sound in gas of the pressure and density which prevail there. On the adiabatic 
hypothesis this gives, by Art. 23 (11), 

*-(£)* <" 

-«-*- i-i^-r- tiyht <»> 

or, if y= 1-408, p = -634p , p=-527p (9) 

If pi be less than this value, the stream after passing the point in question widens out 
again, until it is lost at a distance in the eddies due to viscosity. The minimum sections 
of the elementary streams will be situate in the neighbourhood of the orifice, and their sum 
S may be called the virtual area of the latter. The velocity of efflux, as found from (2), is 

q=-9Uc . 

The rate of discharge is then = qpS, where q and p have the values just found, and is 
therefore approximately independent of the external pressure p x so long as this falls below 
527p . The physical reason of this is (as pointed out by Reynolds) that, so long as the 
velocity at any point exceeds the velocity of sound under the conditions which obtain 
there, no change of pressure can be propagated backwards beyond this point so as to affect 
the motion higher up the stream J. 

Some recent experiments of Stanton § confirm in all essentials the views of Reynolds, 
and clear up some apparent discrepancies. 

Under similar circumstances as to pressure, the velocities of efflux of different gases 
are (so far as y can be assumed to have the same value for each) proportional to the 
corresponding velocities of sound. Hence (as we shall see in Chapter x.) the velocity of 
efflux will vary inversely, and the rate of discharge of mass will vary directly, as the 
square root of the density ||. 

* A result equivalent to this was given by Saint Venant and Wantzel, Journ. de VEcole Polyt. 
xvi. 92 (1839), and was discussed by Stokes, Brit. Ass. Reports for 1846 [Papers, i. 176]. 

f "On the Flow of Gases," Proc. Manch. Lit. and Phil. Soc. Nov. 17, 1885; Phil. Mag. 
March 1886 [Papers, ii. 311]. A similar explanation was given by Hugoniot, Comptes Rendus, 
June 28, July 26, and Dec. 13, 1886. 

X For a further discussion and references see Rayleigh, "On the Discharge of Gases under 
High Pressures," Phil. Mag. (6) xxxii. 177 (1916) [Scientific Papers, Cambridge, 1899-1920, vi. 407]. 

§ Proc. Roy. Soc. A, cxi. 306 (1926). || Cf. Graham, Phil. Trans. 1846. 



28 Integration of the Equations in Special Cases [chap, ii 

Rotating Liquid. 

26. Let us next take the case of a mass of liquid rotating, under the action 
of gravity only, with constant and uniform angular velocity co about the axis 
of z, supposed drawn vertically upwards. 

By hypothesis, u, v, w = — coy, cox, 0, 

X, Y,Z= 0, 0, -g. 

The equation of continuity is satisfied identically, and the dynamical equations 
obviously are 

pdx a pdy pdz 

These have the common integral 

^- = \co 2 (x 2 + y 2 )-gz + const (2) 

P 

The free surface, p = const., is therefore a paraboloid of revolution about the 

axis of z, having its concavity upwards, and its latus rectum = 2g/co 2 . 

«. dv du 
Since - — = 2 co, 

dx cy 

a velocity-potential does not exist. A motion of this kind could not therefore 
be generated in a 'perfect' fluid, i.e. in one unable to sustain tangential stress. 

27. Instead of supposing the angular velocity co to be uniform, let us 
suppose it to be a function of the distance r from the axis, and let us inquire 
what form must be assigned to this function in order that a velocity-potential 
may exist for the motion. We find 

dv du _ 9 dco 

dx dy dr ' 

and in order that this may vanish we must have cor 2 — /x, a constant. The 
velocity at any point is then = /n/r, so that the equation (2) of Art. 21 becomes 



^ = const.-i^, (1) 

p 2 T l V } 



if no extraneous forces act. To find the value of <£ we have, using polar 
co-ordinates, 



dr ' rdO r 



y 



whence cp= — fiO + const. = — /jl tan -1 - + const (2) 

x 

We have here an instance of a 'cyclic' function. A function is said to be 
'single-valued' throughout any region of space when we can assign to every 
point of that region a definite value of the function in such a way that these 
values shall form a continuous system. This is not possible with the function 



26-28] 



Rotating Liquid 



29 



(2) ; for the value of 6, if it vary continuously, changes by — 2irfjb as the point 
to which it refers describes a complete circuit round the origin. The general 
theory of cyclic velocity-potentials will be given in the next chapter. 

If gravity act, and if the axis of z be drawn vertically upwards, we must add 
to (1) the term — gz. The form of the free surface is therefore that generated 
by the revolution of the hyperbolic curve a?z = const, about the axis of z. 

By properly fitting together the two preceding solutions we obtain the case of Rankine's 
' combined vortex.' Thus the motion being everywhere in coaxial circles, let us suppose 
the velocity to be equal to <or from r = to r==a, and to o*a 2 /r for r>a. The corresponding- 
forms of the free surface are then given by 



and 






these being continuous with one another when r=a. The depth of the central depression 
below the general level of the surface is therefore oo 2 a 2 /g. 




28. To illustrate, by way of contrast, the case of extraneous forces not 
having a potential, let us suppose that a mass of liquid filling a right circular 
cylinder moves from rest under the action of the forces 

X = Ax+By, Y=B'x + Cy, Z=0, 

the axis of z being that of the cylinder. 

If we assume u— — coy, v = a>x, w = 0, where o> is a function of t only, these values satisfy 
the equation of continuity and the boundary conditions. The dynamical equations are 
evidently 

<2r= A <r. -X- Tin — , 



dco 

~di 



(d*x=Ax +By 
, 2 y = B'x+Cy- 



■(1) 



1 dp 

dt ™ * ~ " " ' " u p dx ' j 
Differentiating the first of these with respect to y, and the second with respect to x, and 
subtracting, we eliminate p, and find 



%~U*-B). 



■(2) 



30 Integration of the Equations in Special Cases [chap, ii 

The fluid therefore rotates as a whole about the axis of z with constantly accelerated 
angular velocity, except in the particular case when B = B'. To find p, we substitute the 
value of datjdt in (1) and integrate ; we thus get 



p^ 

where 2p=B + B'. 



£o> 2 (x 2 +y 2 ) -f £ {Ax 2 + 2$xy + Cf) + const., 



29. As a final example, we will take one suggested by the theory of 
'electro-magnetic rotations.' 

If an electric current be made to pass radially from an axial wire, through a conducting 
liquid, to the walls of a metallic containing cylinder, in a uniform magnetic field, the 
extraneous forces will be of the type* 

Assuming u= — a>y, v=<ox, w = 0, where to is a function of r and t only, we have 



00) „ uX 1 0» 

3£ ^ r 2 p dy 



Eliminating p, we obtain 2 -=- + r ^-^- = 0. 

The solution of this is a> = F (t)/r 2 +f(r), 

where F and /denote arbitrary functions. If o> = when t=0, we have 

F(0)lr 2 +f(r)=0, 

and therefore o> = „ — — = -* , 



•(2) 



where X is a function of t which vanishes for £=0. Substituting in (1), and integrating, 
we find 



H'-tK'l-^+xw 



Since p is essentially a single-valued function, we must have d\jdt=^ or X=p,t. Hence 
the fluid rotates with an angular velocity which varies inversely as the square of the 
distance from the axis, and increases constantly with the time. 

If C denote the total flux of electricity outwards, per unit length of the axis, and y the 
component of the magnetic force t parallel to the axis, we have u = yCI2irp. The above case is 
specially simple, in that the forces X, Y, Z have a potential (£1= -a tan -1 yjx), though a 'cyclic' 
one. As a rule, in electro-magnetic rotations, this is not the case. 



CHAPTER III 



IRROTATIONAL MOTION 



30. The present chapter is devoted mainly to an exposition of some 
general theorems relating to the kinds of motion already considered in 
Arts. 17-20; viz. those in which udx+ vdy -\-wdz is an exact differential 
throughout a finite mass of fluid. It is convenient to begin with the 
following analysis, due to Stokes*, of the motion of a fluid element in the 
most general case. 

The component velocities at the point (x, y, z) being u, v, w, the relative 
velocities at an infinitely near point (x + Bx, y + By, z + 8z) are 



du * , du z du* 



dx 



dz 



S v = d ^8x + d ^Sy + d £8z, 



dx 



If we write 



a dw g, dw * dw* 

dx dy u dz 

du 
''dx' 



.(1) 



_P_dw dv 
S~dy + dz' 

_ dw dv 
dy dz' 



9 



7 dv 
dy 


dw 
dz 


du dw 

dz dx ' 


, dv du 
dx dy' 



du 

dz 



div 

dx 



£-£. ?= 



dv du 
dx dy 



t. 



(2) 



equations (1) may be written 

Bu — aBx + ^hBy + \gBz + \{rjBz—t > By)/ 

Bv = ihBx+ b8y + if8z + i(Z8x-%8z), - (3) 

8w = | gBx -\- if By + cBz + \{%8y — rjBx). 
Hence the motion of a small element having the point (x, y, z) for its 
centre may be conceived as made up of three parts. 

* " On the Theories of the Internal Friction of Fluids in Motion, &c." Camb. Phil. Trans. 
viii. (1845) [Papers, i, 80]. 

t There is here a deviation from the traditional convention. It has been customary to use 
symbols such as £, 17, £* (Helmholtz) or w', w", a/" (Stokes) to denote the component rotations 
1 /dw dv\ T/du dw\ 1 fdv du\ 
Zydy'TzJ' 2\dz~dx)' 2\dx~dy) 
of a fluid element. The fundamental kinematical theorem is however that of Art. 32 (3), and the 
definition of If, y, f adopted in the text avoids the intrusion of an unnecessary factor 2 (or \ as the 
case may be) in this and in a whole series of subsequent formulae relating to vortex motion. It 
also improves the electro-magnetic analogy of Art. 148. 



32 



Irrotational Motion 



[chap, hi 



The first part, whose components are u, v, w, is a motion of translation of 
the element as a whole. 

The second part, expressed by the first three terms on the right-hand sides 
of the equations (3), is a motion such that, if 8x, 8y y 8z be regarded as current 
co-ordinates, every point is moving in the direction of the normal to that 
quadric of the system 

a {hxf + b (8yf + c (8zf +f8y 8z + g8z8x + h8x8y = const. . . .(4) 
on which it lies. If we refer these quadrics to their principal axes, the cor- 
responding parts of the velocities parallel to these axes will be 

8u' = a'8x', hv' = Vty\ oV = c'S/, (5) 

if a' (oV) 2 + V (8y') 2 + c' {8z'f = const. 

is what (4) becomes by the transformation. The formulae (5) express that 
the length of every line in the element parallel to x' is being elongated at 
the (positive or negative) rate a\ whilst lines parallel to y' and z' are being- 
elongated in like manner at the rates b' and c' respectively. Such a motion is 
called one of pure strain and the principal axes of the quadrics (4) are called 
the axes of the strain. 

The last two terms on the right-hand sides of the equations (3) express a 
rotation of the element as a whole about an instantaneous axis; the component 
angular velocities of the rotation being Jf, ^rj, Jf*. 

The vector whose components are £, 77, f may conveniently be called the 

'vorticity' of the medium at the point (x, y, z). 

This analysis may be illustrated by the so-called 'laminar' motion of a liquid. Thus if 

u = /jLy, y=0, w = 0, 

we have a, b, c,f, g, £, 77 = 0, h = fx, (= -/*. 

If A represent a rectangular fluid element bounded by planes parallel to the co-ordinate 
planes, then B represents the change produced in this in a short time by the strain alone, 
and C that due to the strain plus the rotation. 





It is easily seen that the above resolution of the motion is unique. If we 
assume that the motion relative to the point (x, y, z) can be made up of a 
strain and a rotation in which the axes and coefficients of the strain and the 
axis and angular velocity of the rotation are arbitrary, then calculating the 

* The quantities corresponding to ££, £77, £f in the theory of the infinitely small displacements 
of a continuous medium had been interpreted by Cauchy as expressing the ' mean rotations ' of an 
element, Exercices d' Analyse et de Physique, ii. 302 (Paris, 1841). 



3o-3i] Deformation of an Element 33 

relative velocities Bu, Bv, Bw, we get expressions similar to those on the right- 
hand sides of (3), but with arbitrary values of a, b, c,f, g, h, f, 77, ? Equating 
coefficients of Bx, By, Bz, however, we find that a, b, c, &c. must have respectively 
the same values as before. Hence the directions of the axes of the strain, the 
rates of extension or contraction along them, and the axis and the amount of 
the vorticity, at any point of the fluid, depend only on the state of relative 
motion at that point, and not on the position of the axis of reference. 

When throughout a finite portion of a fluid mass we have f , rj, f all zero, 
the relative motion of any element of that portion consists of a pure strain 
only, and is called 'irrotational.' 

31. The value of the integral 

J(udx + vdy + wdz), 

[( dx dy dz\ 7 
\{ U ds +V Ts* W ds) ds > 

taken along any line ABCD, is called* the 'flow' of the fluid from A to D 
along that line. We shall denote it for shortness by I (ABCD). 

If A and D coincide, so that the line forms a closed curve, or circuit, the 
value of the integral is called the 'circulation' in that circuit. We denote it 
by I (ABC A). If in either case the integration be taken in the opposite 
direction, the signs of dx/ds, dy/ds, dzjds will be reversed, so that we have 

I(AD) = -I(DA), and I (ABCA) = - 1 (ACBA). 
It is also plain that 

I (ABCD) = I (AB) + 1 (BC) 4- 1 (CD). 

Again, any surface may be divided, by a double series of lines crossing 
it, into infinitely small elements. The sum of the circulations round 
the boundaries of these elements, taken all in the 
same sense, is equal to the circulation round the 
original boundary of the surface (supposed for the 
moment to consist of a single closed curve). For, 
in the sum in question, the flow along each side 
common to two elements comes in twice, once for 
each element, but with opposite signs, and there- 
fore disappears from the result. There remain then 
only the flows along those sides which are parts of 
the original boundary; whence the truth of the above 
statement. 

From this it follows, by considerations of continuity, that the circulation 
round the boundary of any surface-element BS, having a given position and 
aspect, is ultimately proportional to the area of the element. 

* Sir W. Thomson, " On Vortex Motion," Edin. Trans, xxv. (1869) [Papers, iv. 13]. 




34 



Irrotational Motion 



[chap, hi 



If the element be a rectangle 8y8z having its centre at the point (a, y, z\ 
then calculating the circulation round it in the direction shewn by the arrows 
in the annexed figure, we have 











y 




\ 


i 


B 






! p 








! A ~ 






B 


1 A 




z 












I{AB) = 
I(CD) = . 



_i 



(dv/dz)8z}8y, I (BC) 



[w + i(dw/dy)8y}8z, 
[w-%(dw/dy)8y}8z, 



and therefore 



[v + i(dv/dz)8z}8y, I (DA)-- 

In this way we infer that the circulations round the boundaries of any 
infinitely small areas 8Si, S$ 2 , 8S 3 , having their planes parallel to the 
co-ordinate planes, are 

fSSi, v8S 2) £8S 3 , (1) 

respectively. 

Again, referring to the figure and the notation of Art. 2, we have 
I {ABC A) = I (PBGP) + / (PGAP) + / (PABP) 

= £ JA + ?; . 77* A + f . nA, 

whence we infer that the circulation round the boundary of any infinitely 
small area 8S is 

(Zf + mi7 + n£)8& (2) 

We have here an independent proof that the quantities f, tj, f, as defined by 
Art. 30 (2), may be regarded as the components of a vector. 

It will be observed that some convention is implied as to the relation 
between the sense in which the circulation round the boundary of 8S is 
estimated, and the sense of the normal (I, m, n). In order to have a clear 
understanding on this point, we shall suppose in this book that the axes of 
co-ordinates form a right-handed system ; thus if the axes of x and y point E. 
and N. respectively, that of z will point vertically upwards*. The sense in 

* Maxwell, Proc. Lond. Math. Soc. (1) iii. 279, 280. Thus in the above diagram the axis of x 
is supposed drawn towards the reader. 



31-33] Circulation in a Finite Circuit 35 

which the circulation, as given by (2), is estimated is then related to the 
direction of the normal (I, m y n) in the manner typified by a right-handed 
screw*. 

32. Expressing now that the circulation round the edge of any finite 
surface is equal to the sum of the circulations round the boundaries of the 
infinitely small elements into which the surface may be divided, we have, 

by (2), 

l(udx + vdy + wdz) = tf(l% + m'n + nZ)dS, (3) 

or, substituting the values of f , 77, f from Art. 30, 

i(^ + ^ + ^)=//f(|4:) +ro s-£) + »g-|)}^... w 

where the single-integral is taken along the bounding curve, and the double- 
integral over the surface f. In these formulae the quantities I, m, n are the 
direction-cosines of the normal drawn always on one side of the surface, which 
we may term the positive side; the direction of integration in the first member 
is then that in which a man walking on the surface, on the positive side of it, 
and close bo the edge, must proceed so as to have the surface always on his 
left hand. 

The theorem (3) or (4) may evidently be extended to a surface whose 
boundary consists of two or more closed curves, provided the integration in 
the first member be taken round each of these in the 

proper direction, according to the rule just given. ^^^^^^^^^^ 

Thus, if the surface-integral in (4) extend over the 
shaded portion of the annexed figure, the directions 
in which the circulations in the several parts of the 
boundary are to be taken are shewn by the arrows, 
the positive side of the surface being that which faces 
the reader. 

The value of the surface-integral taken over a closed surface is zero. 

It should be noticed that (4) is a theorem of pure mathematics, and is 
true whatever functions u, v, w may be of %, y, z, provided only they be con- 
tinuous and differentiable at all points of the surface J. 

33. The rest of this chapter is devoted to a study of the kinematical 
properties of irrotational motion in general, as defined by the equations 

fc%t=0, (1) 

* See Maxwell, Electricity and Magnetism, Oxford, 1873, Art. 23. 

t This theorem is due to Stokes, Smith's Prize Examination Papers for 1854. The first pub- 
lished proof appears to have been given by Hankel, Zur allgem. Theorie der Bewegung der 
Flussigkeiten, Gottingen, 1861. That given above is due to Lord Kelvin, I.e. ante p. 33. See also 
Thomson and Tait, Natural Philosophy, Art. 190 (j), and Maxwell, Electricity and Magnetism, 
Art. 24. 

X It is not necessary that their differential coefficients should be continuous. 




36 Irrotational Motion [chap, hi 

i.e. the circulation in every infinitely small circuit is assumed to be zero. The 
existence and properties of the velocity-potential in the various cases that 
may arise will appear as consequences of this definition. 

The physical importance of the subject rests on the fact that if the 
motion of any portion of a fluid mass be irrotational at any one instant it will 
under certain very general conditions continue to be irrotational. Practically, 
as will be seen, this has already been established by Lagrange's theorem, 
proved in Art. 17, but the importance of the matter warrants a repetition 
of the investigation, in terms of the Eulerian notation, in the form given by 
Lord Kelvin*. 

Consider first any terminated line A B drawn in the fluid, and suppose 
every point of this line to move always with the velocity of the fluid at that 
point. Let us calculate the rate at which the flow along this line, from A to 
B, is increasing. If 8x, 8y, 8z be the projections on the co-ordinate axes of 
an element of the line, we have 

D , k v Da . D8x 

Now D8x/Dt, the rate at which 8x is increasing in consequence of the motion 
of the fluid, is equal to the difference of the velocities parallel to x at the 
two ends of the element, i.e. to 8u\ and the value of Bu/Dt is given by Art. 5. 
Hence, and by similar considerations, we find, if p be a function of p only, 
and if the extraneous forces X, Y, Z have a potential Q, 

T -(u8x + v8y + w8z) = — - — SO + u8u + v8v + w8w. 
JJt p 



Integrating along the line, from A to B, we get 

D [ B 

-y- (udx + vdy + wdz) = 



r 



■(2) 



DtJ A 

or the rate at which the flow from A to B is increasing is equal to the excess 
of the value which — Jdp/p — H + \(f has at B over that which it has at A. 
This theorem comprehends the whole of the dynamics of a perfect fluid. For 
instance, equations (2) of Art. 15 may be derived from it by taking as the 
line AB the infinitely short line whose projections were originally 8a, 8b, 8c, 
and equating separately to zero the coefficients of these infinitesimals. 

If fl be single-valued, the expression within brackets on the right-hand 
side of (2) is a single-valued function of x, y, z. Hence if the integration on 
the left-hand side be taken round a closed curve, so that B coincides with A, 

we have 

D f 

jr I (udx + vdy + wdz) = 0, (3) 

or, the circulation in any circuit moving with the fluid does not alter with 
the time. 

* I.e. ante p. 33. 



33-35] Velocity-Potential 37 

It follows that if the motion of any portion of a fluid mass be initially 
irrotational it will always retain this property; for otherwise the circulation 
in every infinitely small circuit would not continue to be zero, as it is initially 
by virtue of Art. 32 (3). 

34. Considering now any region occupied by irrotationally- moving fluid, 
we see from Art. 32 (3) that the circulation is zero in every circuit which 
can be filled up by a continuous surface lying wholly in the region, or which 
in other words is capable of being contracted to a point without passing out 
of the region. Such a circuit is said to be 'reducible.' 

Again, let us consider two paths A GB, ADB, connecting two points A, B 
of the region, and such that either may by continuous variation be made to 
coincide with the other, without ever passing out of the region. Such paths 
are called 'mutually reconcileable.' Since the circuit AGBDA is reducible, 
we have I (ACBDA)=0, or since I (BDA) = -I(ADB), 

I(ACB) = I(ADB); 

i.e. the flow is the same along any two reconcileable paths. 

A region such that all paths joining any two points of it are mutually 
reconcileable is said to be 'simply-connected.' Such a region is that enclosed 
within a sphere, or that included between two concentric spheres. In what 
follows, as far as Art. 46, we contemplate only simply-connected regions. 

35. The irrotational motion of a fluid within a simply-connected region is 
characterized by the existence of a single-valued velocity-potential. Let us 
denote by — (f> the flow to a variable point P from some fixed point A, viz. 

</> = — I (udx + vdy + wdz) (1) 

J A 

The value of (p has been shewn to be independent of the path along which 
the integration is effected, provided it lie wholly within the region. Hence </> 
is a single-valued function of the position of P; let us suppose it expressed 
in terms of the co-ordinates (a, y, z) of that point. By displacing P through 
an infinitely short space parallel to each of the axes of co-ordinates in 
succession, we find 

u-- d -$ v-- d -& w-- d -i (2) 

i.e. cj> is a velocity-potential, according to the definition of Art. 17. 

The substitution of any other point B for A, as the lower limit of the 
integral in (1), simply adds an arbitrary constant to the value of $, viz. the 
flow from A to B. The original definition of cj> in Art. 17, and the physical 
interpretation in Art. 18, alike leave the function indeterminate to the extent 
of an additive constant. 



38 Irrotational Motion [chap, hi 

As we follow the course of any line of motion the value of <£ continually 
decreases; hence in a simply-connected region the lines of motion cannot 
form closed curves. 

36. The function cf> with which we have here to do is, together with its 
first differential coefficients, by the nature of the case, finite, continuous, and 
single-valued at all points of the region considered. In the case of incom- 
pressible fluids, which we now proceed to consider more particularly, <£ must 
also satisfy the equation of continuity, (6) of Art. 20, or as we shall in future 
write it, for shortness, 

V^ = 0, (1) 

at every point of the region. Hence </> is now subject to mathematical 
conditions identical with those satisfied by the potential of masses attracting 
or repelling according to the law of the inverse square of the distance, at all 
points external to such masses; so that many of the results proved in the 
theories of Attractions, Electrostatics, Magnetism, and the Steady Flow of 
Heat, have also a hydrodynamical application. We proceed to develop those 
which are most important from this point of view. 

In any case of motion of an incompressible fluid the surface-integral of 
the normal velocity taken over any surface, open or closed, is conveniently 
called the 'flux' across the surface. It is of course equal to the volume of fluid 
crossing the surface per unit time. 

When the motion is irrotational, the flux is given by 



-IK* 



where BS is an element of the surface, and 8n an element of the normal to it, 
drawn in the proper direction. In any region occupied wholly by liquid, the 
total flux across the boundary is zero, i.e. 



// 



a,"**- . < 2 > 



the element Sn of the normal being drawn always on one side (say inwards), 
and the integration extending over the whole boundary. This may be regarded 
as a generalized form of the equation of continuity (1). 

The lines of motion drawn through the various points of an infinitesimal 
circuit define a tube, which may be called a tube of flow. The product of 
the velocity (q) into the cross-section (a, say) is the same at all points of such 
a tube. 

We may, if we choose, regard the whole space occupied by the fluid as 
made up of tubes of flow, and suppose the size of the tubes so adjusted that 
the product qa is the same for each. The flux across any surface is then 
proportional to the number of tubes which cross it. If the surface be closed, 



35-38] Tubes of Flow 39 

the equation (2) expresses the fact that as many tubes cross the surface 
inwards as outwards. Hence a line of motion cannot begin or end at a point 
internal to the fluid. 

37. The function $ cannot be a maximum or a minimum at a point in the 
interior of the fluid ; for, if it were, we should have d<f>/dn everywhere positive, 
or everywhere negative, over a small closed surface surrounding the point in 
question. Either of these suppositions is inconsistent with (2). 

Further, the square of the velocity cannot be a maximum at a point 
in the interior of the fluid. For let the axis of x be taken parallel to the 
direction of the velocity at any point P. The equation (1), and therefore also 
the equation (2), is satisfied if we write d<f)/d% for <£. The above argument 
then shews that d<f>/dx cannot be a maximum or a minimum at P. Hence 
there must be points in the immediate neighbourhood of P at which (d(f>/dx) 2 
and therefore a fortiori 

is greater than the square of the velocity at P *. 

On the other hand, the square of the velocity may be a minimum at 
some point of the fluid. The simplest case is that of a zero velocity ; see, for 
example, the figure of Art. 69, below. 

38. Let us apply (2) to the boundary of a finite spherical portion of the 
liquid. If r denote the distance of any point from the centre of the sphere, 
Sot the elementary oolid angle subtended at the centre by an element SS of 
the surface, we have 

d(pldn = — d(f)/dr, 

and &S = r 2 8sx. Omitting the factor r 2 , (2) becomes 



n 



, d«r=0, 

or 



d 

or 



ajj**'- 00 

Since 1/4jt . ffad-n or I/4nrr 2 .jf<f>dS measures the mean value of <f> over 
the surface of the sphere, (3) shews that this mean value is independent of 
the radius. It is therefore the same for any sphere, concentric with the 
former one, which can be made to coincide with it by gradual variation of the 
radius, without ever passing out of the region occupied by the irrotationally 
moving liquid. We may therefore suppose the sphere contracted to a point, 
and so obtain a simple proof of the theorem, first given by Gauss in his 

* This theorem was enunciated, in another connection, by Lord Kelvin, Phil. Mag. Oct. 1850 
[Reprint of Papers on Electrostatics, dtc, London, 1872, Art. 665]. The above demonstration is 
due to Kirchhoff, Vorlesungen iiber mathematische Physik, Mechanik, Leipzig, 1876. For another 
proof see Art. 44 below. 



40 Irrotational Motion [chap, hi 

memoir* on the theory of Attractions, that the mean value of cj> over any 
spherical surface throughout the interior of which (1) is satisfied, is equal to 
its value at the centre. 

The theorem, proved in Art. 37, that <f> cannot be a maximum or a 
minimum at a point in the interior of the fluid, is an obvious consequence of 
the above. 

The above proof appears to be due, in principle, to Frost f. Another demon- 
stration, somewhat different in form, was given by the late Lord RayleighJ. 
The equation (1), being linear, will be satisfied by the arithmetic mean of 
any number of separate solutions fa, fa, fa, — Let us suppose an infinite 
number of systems of rectangular axes to be arranged uniformly about any 
point P as origin, and let fa, fa, fa, ... be the velocity-potentials of motions 
which are the same with respect to these several systems as the original 
motion </> is with respect to the system x, y, z. In this case the arithmetic 
mean (<£, say) of the functions fa, fa, fa, ... will be a function of r, the 
distance from P, only. Expressing that in the motion (if any) represented 
by </>, the flux across any spherical surface which can be contracted to a, point, 
without passing out of the region occupied by the fluid, would be zero, we have 



dr 



or (j) = const. 



39. Again, let us suppose that the region occupied by the irrotationally 
moving fluid is f periphractic,'§ i.e. that it is limited internally by one or 
more closed surfaces, and let us apply (2) to the space included between one 
(or more) of these internal boundaries, and a spherical surface completely 
enclosing it (or them) and lying wholly in the fluid. If M denote the total 
flux into this region, across the internal boundary, we find, with the same 
notation as before, 

80 



I! 



dr 



the surface-integral extending over the sphere only. This may be written 

M 



csll^- 



47T? 



-hence 4^//^ = lr \\^ = £~r + ° (4) 

* " Allgemeine Lehrsatze, u.s.w.," Resultate aus den Beobachtungcn des magnetischenVereins, 
1839 [Werke, Gottingen, 1870-80, v. 199]. 

f Quarterly Journal of Mathematics, xii. (1873). 

X Messenger of Mathematics, vii. 69 (1878) [Papers, i. 347]. 

§ See Maxwell, Electricity and Magnetism, Arts. 18, 22. A region is said to be ' aperiphractic ' 
wlren every closed surface drawn in it can be contracted to a point without passing out of the 
region. 



38-4o] Mean Value over a Spherical Surface 41 

That is, the mean value of <£ over any spherical surface drawn under the 
above-mentioned conditions is equal to Mj^mrr -I- G, where r is the radius, M 
an absolute constant, and G a quantity which is independent of the radius 
but may vary with the position of the centre*. 

If however the original region throughout which the irrotational motion 
holds be unlimited externally, and if the first derivative (and therefore all the 
higher derivatives) of c/> vanish at infinity, then G is the same for all spherical 
surfaces enclosing the whole of the internal boundaries. For if such a sphere 
be displaced parallel to xf, without alteration of size, the rate at which C 
varies in consequence of this displacement is, by (4), equal to the mean value 
of d(j>/dx over the surface. Since dcp/dx vanishes at infinity, we can by taking 
the sphere large enough make the latter mean value as small as we please. 
Hence C is not altered by a displacement of the centre of the sphere parallel 
to x. In the same way we see that G is not altered by a displacement parallel 
to y or z\ i.e. it is absolutely constant. 

If the internal boundaries of the region considered be such that the total 
flux across them is zero, e.g. if they be the surfaces of solids, or of portions of 
incompressible fluid whose motion is rotational, we have M = 0, so that the 
mean value of (/> over any spherical surface enclosing them all is the same. 

40. (a) If cf> be constant over the boundary of any simply-connected 
region occupied by liquid moving irrotationally, it has the same constant 
value throughout the interior of that region. For if not constant it would 
necessarily have a maximum or a minimum value at some point of the region. 

Otherwise: we have seen in Arts. 35, 36 that the lines of motion cannot 
begin or end at any point of the region, and that they cannot form closed 
curves lying wholly within it. They must therefore traverse the region, 
beginning and ending on its boundary. In our case however this is impossible, 
for such a line always proceeds from places where </> is greater to places where 
it is less. Hence there can be no motion, i.e. 

dcf> d<j> d(f) _ « 
dx ' dy ' dz 

and therefore cj> is constant and equal to its value at the boundary. 

(/3) Again, if d</>/dn be zero at every point of the boundary of such a 
region as is above described, <j> will be constant throughout the interior. For 
the condition d<p/dn = expresses that no lines of motion enter or leave the 
region, but that they are all contained within it. This is however, as we have 
seen, inconsistent with the other conditions which the lines must conform 
to. Hence, as before, there can be no motion, and cj> is constant. 

* It is understood, of course, that the spherical surfaces to which this statement applies are 
reconcileable with one another, in a sense analogous to that of Art. 34. 
f Kirchhoff, Mechanik, p. 191. 



42 Irrotational Motion [chap, hi 

This theorem may be otherwise stated as follows: no continuous irrota- 
tional motion of a liquid can take place in a simply-connected region bounded 
entirely by fixed rigid walls. 

(7) Again, let the boundary of the region considered consist partly of 
surfaces S over which <f> has a given constant value, and partly of other 
surfaces 2 over which d<f>/dn = 0. By the previous argument, no lines of 
motion can pass from one point to another of S, and none can cross S. Hence 
no such lines exist ; <p> is therefore constant as before, and equal to its value 
at 8. 

It follows from these theorems that the irrotational motion of a liquid in 
a simply-connected region is determined when either the value of <f>, or the 
value of the inward normal velocity — d<f>/dn, is prescribed at all points of the 
boundary, or (again) when the value of cf> is given over part of the boundary, 
and the value of — d<j>/dn over the remainder. For if fa , fa be the velocity- 
potentials of two motions each of which satisfies the prescribed boundary- 
conditions, in any one of these cases, the function fa — fa satisfies the condition 
(a) or (/3) or (7) of the present Article, and must therefore be constant 
throughout the region. 

41. A class of cases of great importance, but not strictly included in the 
scope of the foregoing theorems, occurs when the region occupied by the 
irrotationally moving liquid extends to infinity, but is bounded internally by 
one or more closed surfaces. We assume, for the present, that this region is 
simply-connected, and that (j> is therefore single-valued. 

If cj> be constant over the internal boundary of the region, and tend every- 
where to the same constant value at an infinite distance from the internal 
boundary, it is constant throughout the region. For otherwise <£ would be a 
maximum or a minimum at some point within the region. 

We infer, exactly as in Art. 40, that if <f> be given arbitrarily over the 
internal boundary, and have a given constant value at infinity, its value is 
everywhere determinate. 

Of more importance in our present subject is the theorem that, if the 
normal velocity be zero at every point of the internal boundary, and if the 
fluid be at rest at infinity, then </> is everywhere constant. We cannot how- 
ever infer this at once from the proof of the corresponding theorem in Art. 40. 
It is true that we may suppose the region limited externally by an infinitely 
large surface at every point of which d<j>/dn is infinitely small; but it is 
conceivable that the integral ffd<p/dn . dS, taken over a portion of this surface, 
might still be finite, in which case the investigation referred to would fail. 
We proceed therefore as follows. 

Since the velocity tends to the limit zero at an infinite distance from the 
internal boundary (S, say), it must be possible to draw a closed surface 2 



40-42] Conditions of Determinateness 43 

completely enclosing S, beyond which the velocity is everywhere less than a 
certain value e, which value may, by making 2 large enough, be made as 
small as we please. Now in any direction from S let us take a point P at 
such a distance beyond 2 that the solid angle which 2 s.ubtends at it is 
infinitely small; and with P as centre let us describe two spheres, one just 
excluding, the other just including S. We shall prove that the mean value 
of (j> over each of these spheres is, within an infinitely small amount, the 
same. For if Q, Q' be points of these spheres on a common radius PQQ' , then 
if Q, Q' fall within 2 the corresponding values of (f> may differ by a finite 
amount; but since the portion of either spherical surface which falls within X 
is an infinitely small fraction of the whole, no finite difference in the mean 
values can arise from this cause. On the other hand, when Q, Q' fall without 
%, the corresponding values of <£ cannot differ by so much as e . QQ', for e is 
by definition a superior limit to the rate of variation of <£. Hence, the mean 
values of <f> over the two spherical surfaces must differ by less than e . QQ' . 
Since QQ' is finite, whilst e may by taking S large enough be made as small 
as we please, the difference of the mean values may, by taking P sufficiently 
distant, be made infinitely small. 

Now we have seen in Arts. 38, 39 that the mean value of </> over the inner 
sphere is equal to its value at P, and that the mean value over the outer 
sphere is (since M = 0) equal to a constant quantity C. Hence, ultimately, the 
value of <f> at infinity tends everywhere to the constant value C. 

The same result holds even if the normal velocity be not zero over the 
internal boundary; for in the theorem of Art. 39 M is divided by r, which is 
in our case infinite. 

It follows that if d(f)/dn = at all points of the internal boundary, and if 
the fluid be at rest at infinity, it must be everywhere at rest. For no lines 
of motion can begin or end on the internal boundary. Hence such lines, if 
they existed, must come from an infinite distance, traverse the region occupied 
by the fluid, and pass off again to infinity; i.e. they must form infinitely long 
courses between places where <£ has, with an infinitely small amount, the 
same value (7, which is impossible. 

The theorem that, if the fluid be at rest at infinity, the motion is deter- 
minate when the value of — d(f>/dn is given over the internal boundary, follows 
by the same argument as in Art. 40. 

Greens Theorem. 

42. In treatises on Electrostatics, &c, many important properties of the 
potential are usually proved by means of a certain theorem due to Green. Of 
these the most important from our present point of view have already been 
given; but as the theorem in question leads, amongst other things, to a useful 



44 Irrotational Motion [chap, hi 

expression for the kinetic energy in any case of irrotational motion, some 
account of it will properly find a place here. 

Let U, V, W be any three functions which are finite, single-valued and 
differentiable at all points of a connected region completely bounded by one 
or more closed surfaces S; let 88 be an element of any one of these surfaces, 
and I, m, n the direction-cosines of the normals to it drawn inwards. We shall 
prove in the first place that 

[l(W + mV+nW)dS=-lj((^+ d ^ (1) 

where the triple-integral is taken throughout the region, and the double- 
integral over its boundary. 

Tf we conceive a series of surfaces drawn so as to divide the region into 
any number of separate parts, the integral 

fj(lU + mV + nW)dS, (2) 

taken over the original boundary, is equal to the sum of the similar integrals 
each taken over the whole boundary of one of these parts. For, for every 
element. 8a of a dividing surface, we have, in the integrals corresponding to 
the parts lying on the two sides of this surface, elements (IU + mV+nW) 8a, 
and (I'U + m'V + n W) 8a, respectively. But the normals to which I, m, n 
and V , m ', n refer being drawn inwards in each case, we have V = — I, m! = — m, 
n' = — n; so that, in forming the sum of the integrals spoken of, the elements 
due to the dividing surfaces disappear, and we have left only those due to the 
original boundary of the region. 

Now let us suppose the dividing surfaces to consist of three systems of 
planes, drawn at infinitesimal intervals, parallel to yz, zx, xy, respectively. If 
x, y, z be the co-ordinates of the centre of one of the rectangular spaces thus 
formed, and 8x, 8y, 8z the lengths of its edges, the part of the integral (2) due 
to the yz-f&ce nearest the origin is 

and that due to the opposite face is 

-(u + i^Sa^SyBz. 

The sum of these is — dU/dx . 8x8y8z. Calculating in the same way the parts 
of the integral due to the remaining pairs of faces, we get for the final result 

Hence (1) simply expresses the fact that the surface-integral (2), taken over 
the boundary of the region, is equal to the sum of the similar integrals taken 



42-43] Green's Theorem 45 

over the boundaries of the elementary spaces of which we have supposed it 
built up. 

It is evident from (1), or it may be proved directly by transformation of 
co-ordinates, that if U, V, W be regarded as components of a vector, the 
expression 

d_u dv dw 

dx dy dz 

is a 'scalar' quantity, i.e. its value is unaffected by any such transformation. 
It is now usually called the 'divergence' of the vector-field at the point (x, y, z). 

The interpretation of (1), when (U, V, W) is the velocity of a continuous 
substance, is obvious. In the particular case of irrotational motion we obtain 

d ^dS = -ljlv*<j>dxdydz, (3) 

where 8n denotes an element of the inwardly-directed normal to the surface S. 

Again, if we put U, V, W = pu, pv, pw, respectively, we reproduce in 
substance the second investigation of Art. 7. 

Another useful result is obtained by putting U, V, W = i«/>, v<fi, wcf>, respec- 
tively, where u, v, w satisfy the relation 

du dv dw _ 
dx dy dz 

throughout the region, and make 

lu i-mv + nw = 
over the boundary. We find 



ii 



ll!{%+'%+'%)**?-o <*> 



dy 

The function cj> is here merely restricted to be finite, single- valued, and con- 
tinuous, and to have its first differential coefficients finite, throughout the 
region. 

43. Now let 0, (/>' be any two functions which, together with their first 
and second derivatives, are finite and single-valued throughout the region 
considered; and let us put 

d(h' 
respectively, so that IU + mV + n W =</>•—. 

Substituting in (1) we find 

-!J!<tX*<t>'dxdydz ..(5) 



46 Irrotational Motion [chap, hi 

By interchanging </> and <fi we obtain 

-$JS$V 2 4>dxdydz (6) 

Equations (5) and (6) together constitute Green's theorem* 

44. If <£, (/>' be the velocity-potentials of two distinct modes of irrotational 
motion of a liquid, so that 

V 2 </> = 0, V 2 </>' = 0, (1) 

we obtain jf^dS- jfo'^dS (2) 

If we recall the physical interpretation of the velocity-potential, given in 
Art. 18, then, regarding the motion as generated in each case impulsively 
from rest, we recognize this equation as a particular case of the dynamical 
theorem that 

tp r qr =Zpr'qr, 

where p r} q r and p r ', q r ' are generalized components of impulse and velocity, 
in any two possible motions of a system f. 

Again, in Art. 43 (6) let </>' = <\>, and let <f> be the velocity-potential of a 
liquid. We obtain 

(^♦gr*«n***~j«« <»> 

To interpret this we multiply both sides by ^p. Then on the right-hand 
side — d(j>/dn denotes the normal velocity of the fluid inwards, whilst p<\> is, by 
Art. 18, the impulsive pressure necessary to generate the motion. It is a 
proposition in Dynamics J that the work done by an impulse is measured by 
the product of the impulse into half the sum of the initial and final velocities, 
resolved in the direction of the impulse, of the point to which it is applied. 
Hence the right-hand side of (3), when modified as described, expresses the 
work done by the system of impulsive pressures which, applied to the surface 
S, would generate the actual motion; whilst the left-hand side gives the 
kinetic energy of this motion. The formula asserts that these two quantities 
are equal. Hence if T denote the total kinetic energy of the liquid, we have 
the very important formula 

tT — pjj^d8. (4) 

If in (3), in place of <f), we write d<f)/dx, which will of course satisfy V 2 3$/9#=0, and 
ipply the resulting theorem to the region included within a spherical surface of radius r 

* G. Green, Essay on Electricity and Magnetism, Nottingham, 1828, Art. 3 [Mathematical 
Papers (ed. Ferrers), Cambridge, 1871, p. 3]. 

f Thomson and Tait, Natural Philosophy, Art. 313, equation (11). 
X Ibid. Art. 308. 



43-45] Kinetic Energy 47 

having any point (x, y, z) as centre, then with the same notation as in Art. 39, we have 



Hence, writing q 2 = u 2 + v 2 + iv 2 , 

Since this latter expression is essentially positive, the mean value of q 2 , taken over a 
sphere having any given point as centre, increases with the radius of the sphere. Hence 
q 2 cannot be a maximum at any point of the fluid, as was proved otherwise in Art. 37. 

Moreover, recalling the formula for the pressure in any case of irrotational motion of a 
liquid, viz. 

&J*+-a-W+F{t\ (6) 

we infer that, provided the potential Q. of the external forces satisfy the condition 

V 2 S2=0, (7) 

the mean value of p over a sphere described with any point in the interior of the fluid as 
centre will diminish as the radius increases. The place of least pressure will therefore be 
somewhere on the boundary of the fluid. This has a bearing on the point discussed in 
Art. 23. 



45. In this connection we may note a remarkable theorem discovered by 
Lord Kelvin*, and afterwards generalized by him into an universal property 
of dynamical systems started impulsively from rest under prescribed velocity- 
conditions f. 

The irrotational motion of a liquid occupying a simply-connected region 
has less kinetic energy than any other motion consistent with the same normal 
motion of the boundary. 

Let The the kinetic energy of the irrotational motion to which the velocity- 
potential $ refers, and T± that of another motion given by 

dd> dd> dd> , n \ 

U = -dx +U< " V = -frj +V °' W = -dI + W °' (8) 

where, in virtue of the equation of continuity, and the prescribed boundary- 
condition, we must have 

duo dvo dwo _ 
dx dy dz 

throughout the region, and lu + mv + nw = 
over the boundary. Further let us write 

To = ip!f!(u<? + v 2 + w 2 )dxdydz (9) 

* (W. Thomson) "On the Vis-Viva of a Liquid in Motion," Camb. and Dub. Math. Journ. 
1849 [Papers, i. 107]. 

t Thomson and Tait, Art. 312. 



48 Irrotational Motion [chap, hi 

We find T 1 = T+T - pj'JfUo^ + v ^ + w ^ dxdydz. 

Since the last integral vanishes, by Art. 42 (4), we have 

r l\ = T+T Qy (10) 

which proves the theorem*. 

46. We shall require to know, hereafter, the form assumed by the ex- 
pression (4) for the kinetic energy when the fluid extends to infinity and is 
at rest there, being limited internally by one or more closed surfaces S. Let 
us suppose a large closed surface X described so as to enclose the whole of S. 
The energy of the fluid included between S and % is 

-*,//♦£*-!,//$.* (ID 

where the integration in the first term extends over S, that in the second over 
2. Since we have, by the equation of continuity, 

the expression (11) may be written 

-Ipffa-O^dS-lpfjw-C^dl, (12) 

where G may be any constant, but is here supposed to be the constant value 
to which <j) was shewn in Art. 39 to tend at an infinite distance from S. Now 
the whole region occupied by the fluid may be supposed made up of tubes of 
flow, each of which must pass either from one point of the internal boundary 
to another, or from that boundary to infinity. Hence the value of the integral 

d<j> 



si 



dn d1 ' 



taken over any surface, open or closed, finite or infinite, drawn within the 
region, must be finite. Hence ultimately, when 2 is taken infinitely large and 
infinitely distant all round from S, the second term of (12) vanishes, and we 
have 

2T = -pjj(<p-C) d ^dS, (13) 

where the integration extends over the internal boundary only. 
If the total flux across the internal boundarv be zero, we have 



ff 



d ^dS = 0, 

on 



so that (13) may be written 2T = - p jU^dS, (14) 

simply. 

* Some extensions of this result are discussed by Leathern, Cambridge Tracts, No. 1, 2nd ed. 
(1913). They supply further interesting illustrations of Kelvin's general dynamical principle. 



45-47] Cyclic Regions 49 

On Multiply-connected Regions. 

47. Before discussing the properties of irrotational motion in multiply- 
connected regions we must examine more in detail the nature and classifica- 
tion of such regions. In the following synopsis of this branch of the geometry 
of position we recapitulate for the sake of completeness one or two definitions 
already given. 

We consider any connected region of space, enclosed by boundaries. A 
region is 'connected' when it is possible to pass from any one point of it to any 
other by an infinity of paths, each of which lies wholly in the region. 

Any two such paths, or any two circuits, which can by continuous variation 
be made to coincide without ever passing out of the region, are said to be 
'mutually reconcileable.' Any circuit which can be contracted to a point 
without passing out of the region is said to be 'reducible/ Two reconcileable 
paths, combined, form a reducible circuit. If two paths or two circuits be 
reconcileable, it must be possible to connect them by a continuous surface, 
which lies wholly within the region, and of which they form the complete 
boundary: and conversely. 

It is further convenient to distinguish between 'simple' and 'multiple' 
irreducible circuits. A 'multiple' circuit is one which cau by continuous 
variation be made to appear, in whole or in part, as the repetition of another 
circuit a certain number of times. A 'simple' circuit is one with which this 
is not possible. 

A 'barrier,' or 'diaphragm,' is a surface drawn across the region, and 
limited by the line or lines in which it meets the boundary. Hence a barrier 
is necessarily a connected surface, and cannot consist of two or more detached 
portions. 

A 'simply-connected' region is one such that all paths joining any two 
points of it are reconcileable, or such that all circuits drawn within it are 
reducible. 

A 'doubly-connected' region is one such that two irreconcileable paths, 
and no more, can be drawn between any two points A, B of it; viz. any other 
path joining AB is reconcileable with one of these, or with a combination of 
the two taken each a certain number of times. In other words, the region is 
such that one (simple) irreducible circuit can be drawn in it, whilst all other 
circuits are either reconcileable with this (repeated, if necessary), or are 
reducible. As an example of a doubly-connected region we may take that 
enclosed by the surface of an anchor-ring, or that external to such a ring and 
extending to infinity. 

Generally, a region such that n irreconcileable paths, and no more, can be 
drawn between any two points of it, or such that n — 1 (simple) irreducible 



50 Irrotational Motion [chap, hi 

and irreconcileable circuits, and no more, can be drawn in it, is said to be 
'^-ply-connected.' 

The shaded portion of the figure on p. 35 is a triply-connected space of 
two dimensions. 

It may be shewn that the above definition of an w-ply-connected space is 
self-consistent. In such simple cases as n = 2, n = 3, this is sufficiently evident 
without demonstration. 

48. Let us suppose, now, that w 7 e have an w-ply-connected region, with 
n — 1 simple independent irreducible circuits drawn in it. It is possible to 
draw a barrier meeting any one of these circuits in one point only, and not 
meeting any of the n — 2 remaining circuits. A barrier drawn in this manner 
does not destroy the continuity of the region, for the interrupted circuit remains 
as a path leading round from one side to the other. The order of connection 
of the region is however diminished by unity; for every circuit drawn in the 
modified region must be reconcileable with one or more of the n — 2 circuits 
not met by the barrier. 

A second barrier, drawn in the same manner, will reduce the order of con- 
nection again by one, and so on ; so that by drawing n — 1 barriers we can 
reduce the region to a simply-connected one. 

A simply-connected region is divided by a barrier into two separate 
parts; for otherwise it would be possible to pass from a point on one side 
of the barrier to an adjacent point on the other side by a path lying wholly 
within the region, which path would in the original region form an irreducible 
circuit. 

Hence in an w-ply-connected region it is possible to draw n — 1 barriers, 
and no more, without destroying the continuity of the region. This property 
is sometimes adopted as the definition of an ^-ply-connected space. 

Irrotational Motion in Multiply -connected Spaces. 

49. The circulation is the same in any two reconcileable circuits ABC A, 
A'B'G'A' drawn in a region occupied by fluid moving irrotationally. For the 
two circuits may be connected by a continuous surface lying wholly within 
the region; and if we apply the theorem of Art. 32 to this surface, we 
have, remembering the rule as to the direction of integration round the 
boundary. 

I(ABCA) + I(A'C'B'A') = 0, 
or 1 (ABGA) = I (A'B'C'A'). 

If a circuit ABGA be reconcileable with two or more circuits A'B'C A' , 
A"B"G" A" , &c, combined, we can connect all these circuits by a continuous 
surface which lies wholly within the region, and of which they form the com- 
plete boundary. Hence 

J (ABGA) + 1 (A'G'B'A') + / (A"G"B"A") + &c. = 0, 
or I{ABGA) = I{A'B , C'A , ) + I(A"B"G"A") + &z.; 



47-50] Cyclic Velocity -Potentials 51 

i.e. the circulation in any circuit is equal to the sum of the circulations in the 
several members of any set of circuits with which it is reconcileable. 

Let the order of connection of the region be n + 1, so that n independent 
simple irreducible circuits a 1} a 2 , ... a n can be drawn in it; and let the circu- 
lations in these be k 1} k 2 , ... fc n , respectively. The sign of any k will of course 
depend on the direction of integration round the corresponding circuit ; let 
the direction in which k is estimated be called the positive direction in the 
circuit. The value of the circulation in any other circuit can now be found 
at once. For the given circuit is necessarily reconcileable with some com- 
bination of the circuits a lt a 2 , ... a n ; say with a x taken p ± times, a 2 taken 
p 2 times and so on, where of course any p is negative when the corre- 
sponding circuit is taken in the negative direction. The required circulation 
then is 

PifC!-{-p 2 /c 2 + ...+p n K n (1) 

Since any two paths joining two points A, B of the region together form 
a circuit, it follows that the values of the flow in the two paths differ by 
a quantity of the form (1), where, of course, in particular cases some or all of 
the p's may be zero. 

50. Let us denote by — <p the flow to a variable point P from a fixed 
point A, viz. 

(/> = — (udcc + vdy + wdz) (2) 

J A 

So long as the path of integration from A to P is not specified, </> is indeter- 
minate to the extent of a quantity of the form (1). 

If however n barriers be drawn in the manner explained in Art. 48, so as 
to reduce the region to a simply-connected one, and if the path of integration 
in (2) be restricted to lie within the region as thus modified {i.e. it is not to 
cross any of the barriers), then <f> becomes a single-valued function, as in 
Art. 35. It is continuous throughout the modified region, but its values at 
two adjacent points on opposite sides of a barrier differ by ± k. To derive the 
value of (j) when the integration is taken along any path in the unmodified 
region we must subtract the quantity (1), where any p denotes the number of 
times this path crosses the corresponding barrier. A crossing in the positive 
direction of the circuits interrupted by the barrier is here counted as positive, 
a crossing in the opposite direction as negative. 

By displacing P through an infinitely short space parallel to each co-ordinate 
axis in succession, we find 

dd> dd> deb 

U ' V > W = ~£' ~Jy- ~Tz> 

so that </> satisfies the definition of a velocity- potential (Art. 17). It is now 

however a many-valued or cyclic function ; i.e. it is not possible to assign to 

every point of the original region a unique and definite value of <£, such values 



/: 




52 Irrotational Motion [chap, in 

forming a continuous system. On the contrary, whenever P describes an irre- 
ducible circuit, </> will not, in general, return to its original value, but will differ 
from it by a quantity of the form (1). The quantities *i, k 2 , ... /c n , which specify 
the amounts by which <£ decreases as P describes the several independent 
circuits of the region, may be called the 'cyclic constants' of </>. 

It is an immediate consequence of the 'circulation-theorem' of Art. 33 
that under the conditions there presupposed the cyclic constants do not alter 
with the time. The necessity for these conditions is exemplified in the problem 
of Art. 29, where the potential of the extraneous forces is itself a cyclic 
function. 

The foregoing theory may be illustrated by the case of Art. 27 (2), where the region (as 
limited by the exclusion of the origin, since the formula would give an infinite velocity there) 
is doubly-connected ; for we can connect any two points A, B of it by two irreconcileable 
paths passing on opposite sides of the axis of 2, e.g. 
ACB, ADB in the figure. The portion of the plane zx 
for which x is positive, may be taken as a barrier, 
and the region is thus made simply-connected. The 
circulation in any circuit meeting this barrier once 
only, e.g. in ACB DA, is 

•2tt 

fijr. rd&, or 27r/x. 

That in any circuit not meeting the barrier is zero. In the modified region <£ may be put 
equal to a single- valued function, viz. — fid, but its value on the positive side of the barrier 
is zero, that at an adjacent point on the negative is — 2-rrfi. 

More complex illustrations of irrotational motion in multiply-connected spaces of two 
dimensions will present themselves in the next chapter. 

51. Before proceeding further we may briefly indicate a somewhat different 
method of presenting the above theory. 

Starting from the existence of a velocity-potential as the characteristic 
of the class of motions which we propose to study, and adopting the second 
definition of an n+ 1 -ply-connected region, indicated in Art. 48, we remark 
that in a simply-connected region every equipotential surface must either be 
a closed surface, or else form a barrier dividing the region into two separate 
parts. Hence, supposing the whole system of such surfaces drawn, we see that 
if a closed curve cross any given equipotential surface once it must cross it 
again, and in the opposite direction. Hence, corresponding to any element of 
the curve, included between two consecutive equipotential surfaces, we have a 
second element such that the flow along it, being equal to the difference 
between the corresponding values of </>, is equal and opposite to that along the 
former; so that the circulation in the whole circuit is zero. 

If however the region be multiply-connected, an equipotential surface 
may form a barrier without dividing it into two separate parts. Let as 
many such surfaces be drawn as is possible without destroying the con- 
tinuity of the region. The number of these cannot, by definition, be greater 



50-52] Multiple Connectivity 53 

than n. Every other equipotential surface which is not closed will be re- 
concileable (in an obvious sense) with one or more of these barriers. A curve 
drawn from one side of a barrier round to the other, without meeting any of 
the remaining barriers, will cross every equipotential surface reconcileable 
with the first barrier an odd number of times, and every other equipotential 
surface an even number of times. Hence the circulation in the circuit thus 
formed will not vanish, and </> will be a cyclic function. 

In the method adopted above we have based the whole theory on the 
equations 

dw dv _ n du dw _ d_v _ du _ , . 

dy dz ' dz dx ' dx dy 

and have deduced the existence and properties of the velocity-potential in 
the various cases as necessary consequences of these. In fact, Arts. 34. 35, 
and 49, 50 may be regarded as an inquiry into the nature of the solution of 
this system of differential equations, as depending on the character of the 
region through which they hold. 

The integration of (3), when we have, on the right-hand side, instead of 
zero, known functions of x, y, z, will be treated in Chapter vn. 

52. Proceeding now, as in Art. 36, to the particular case of an incom- 
pressible fluid, we remark that whether <f> be cyclic or not, its first derivatives 
d(f>/dx, d(j>/dy, d4>/dz, and therefore all the higher derivatives, are essentially 
single-valued functions, so that <j> will still satisfy the equation of continuity 

V 2 = O, (1) 

or the equivalent form II ~ dS — 0, (2) 

where the surface-integration extends over the whole boundary of any portion 
of the fluid. 

The theorem (a) of Art. 40, viz. that <£ must be constant throughout the 
interior of any region at every point of which (1) is satisfied, if it be constant 
over the boundary, still holds when the region is multiply-connected. For <£, 
being constant over the boundary, is necessarily single-valued. 

The remaining theorems of Art. 40, being based on the assumption that 
the stream-lines cannot form closed curves, will require modification. We 
must introduce the additional condition that the circulation is to be zero in 
each circuit of the region. 

Removing this restriction, we have the theorem that the irrotational 
motion of a liquid occupying an n-ply-connected region is determinate when 
the normal velocity at every point of the boundary is prescribed, as well as 
the value of the circulation in each of the n independent and irreducible 
circuits which can be drawn in the region. For if <j> 1} <£ 2 be the (cyclic) 
velocity-potentials of two motions satisfying the above conditions, then 



54 Irrotational Motion [chap, hi 

<£ = <£i — $2 is a single- valued function which satisfies (1) at every point of 
the region, and makes d<f)/dn = at every point of the boundary. Hence, by 
Art. 40, cf) is constant, and the motions determined by </>i and </> 2 are identical. 

The theory of multiple connectivity seems to have been first developed by Riemann* 
for spaces of two dimensions, a propos of his researches on the theory of functions of a 
complex variable, in which connection also cyclic functions satisfying the equations 

da 2 dy' 2 
through multiply-connected regions, present themselves. 

The bearing of the theory on Hydrodynamics and the existence in certain cases of 
many-valued velocity-potentials were first pointed out by von Helmholtzt. The subject 
of cyclic irrotational motion in multiply-connected regions was afterwards taken up and 
fully investigated by Lord Kelvin in the paper on vortex-motion already referred to J. 

Kelvins Extension of Greens Theorem. 

53. It was assumed in the proof of Green's theorem that </> and <f>' were 
both single-valued functions. If either be a cyclic function, as may be the 
case when the region to which the integrations in Art. 43 refer is multiply- 
connected, the statement of the theorem must be modified. Let us suppose, 
for instance, that cj> is cyclic; the surface-integral on the left-hand side of 
Art. 43 (5), and the second volume-integral on the right-hand side, are then 
indeterminate, on account of the indeterminateness in the value of <£ itself. 
To remove this indeterminateness, let the barriers necessary to reduce the 
region to a simply-connected one be drawn, as explained in Art. 48. We 
may now suppose <fi to be continuous and single-valued throughout the region 
thus modified ; and the equation referred to will then hold, provided the two 
sides of each barrier be reckoned as part of the boundary of the region, and 
therefore included in the surface-integral on the left-hand side. Let oVi, 
be an element of one of the barriers, kx the cyclic constant corresponding to 
that barrier, d<f>'/dn the rate of variation of <£' in the positive direction of the 
normal to 8<ri'. Since, in the parts of the surface-integral due to the two 
sides of So"!, d<j>'/dn is to be taken with opposite signs, whilst the value of </> 
on the positive side exceeds that on the negative side by tci, we get finally 
for the element of the integral due to So"i, the value /c^fi/dn. Bai. Hence 
Art. 43 (5) becomes, in the altered circumstances, 

* Grundlagen fiir eine allgemeine Theorie der Funetionen einer veranderlichen complexen 
Gr'osse, Gottingen, 1851 [Matheviatische Werke, Leipzig, 1876, p. 3]. Also: "Lehrsatze aus der 
Analysis Situs," Crelle, liv. (1857) [Werke, p. 84]. t Crelle, lv. (1858). 

J See also Kirchhoff , ' ' Ueber die Krai te welche zwei unendlich diinne starre Hinge in einer 
Fliissigkeit scheinbar auf einander ausiiben konnen," Crelle, lxxi. (1869) [Gesammelte Abhand- 
lungen, Leipzig, 1882, p. 404]. 



52-54] Extension of Green's Theorem 55 

where the surface-integrations indicated on the left-hand side extend, the 
first over the original boundary of the region only, and the rest over the 
several barriers. The coefficient of any k is evidently minus the total flux 
across the corresponding barrier, in a motion of which <f>' is the velocity- 
potential. The values of <j> in the first and last terms of the equation are to 
be assigned in the manner indicated in Art. 50. 

If 4>' also be a cyclic function, having the cyclic constants «/, k 2 , &c., 
then Art. 43 (6) becomes in the same way 



h't^A\t^A\l 



da 2 + • . . 



Equations (1) and (2) together constitute Lord Kelvin's extension of Green's 
theorem. 

54. If <f), ()>' are both velocity-potentials of a liquid, we have 

V 2 (/> = 0, V 2 f = 0, (3) 

and therefore <£ ■— dS + K\ \\-^- do-} + k 2 -^- da 2 + . . . 

-JJ'S^JS^Wf^ (4 > 

To obtain a physical interpretation of this theorem it is necessary to 
explain in the first place a method, imagined by Lord Kelvin, of generating 
any given cyclic irrotational motion of a liquid "in a multiply-connected 
space. 

Let us suppose the fluid to be enclosed in a perfectly smooth and flexible 
membrane occupying the position of the boundary. Further, let n barriers 
be drawn, as in Art. 48, so as to convert the region into a simply-connected 
one, and let their places be occupied by similar membranes, infinitely thin, 
and destitute of inertia. The fluid being initially at rest, let each element 
of the first-mentioned membrane be suddenly moved inwards with the given 
(positive or negative) normal velocity — d(f>/dn, whilst uniform impulsive 
pressures Kip,/c 2 p,... K n p are simultaneously applied to the negative sides of 
the respective barrier-membranes. The motion generated will be characterized 
by the following properties. It will be irrotational, being generated from 
rest ; the normal velocity at every point of the original boundary will have 
the prescribed value ; the values of the impulsive pressure at two adjacent 
points on opposite sides of a membrane will differ by the corresponding value 
of Kp, and the values of the velocity-potential will therefore differ by the 
corresponding value of k ; finally, the motion on one side of a barrier will be 
continuous with that on the other. To prove the last statement we remark, 
first, that the velocities normal to the barrier at two adjacent points on 



56 Irrotational Motion [chap, hi 

opposite sides of it are the same, being each equal to the normal velocity of 
the adjacent portion of the membrane. Again, if P, Q be two consecutive 
points on a barrier, and if the corresponding values of <£ be on the positive 
side </> P , (pQ, and on the negative side $ V P , <£ v q, we have 

and therefore (f> Q — cj> P — $ V Q — <£ V P , 

i.e., if PQ = Bs, d(f>/ds = dfi/ds. 

Hence the tangential velocities at two adjacent points on opposite sides of 
the barrier also agree. If then we suppose the barrier-membranes to be 
liquefied immediately after the impulse, we obtain the irrotational motion 
in question. 

The physical interpretation of (4), when multiplied by — p, now follows 
as in Art. 44. The values of p/c are additional components of momentum, 
and those of —ffd<f>/dn.d(r, the fluxes through the various apertures of the 
region, are the corresponding generalized velocities. 

55. If in (2) we put (f> r = <j>, and suppose $ to be the velocity-potential of 
an incompressible fluid, we find 

= - p jj'i >d £ ds - p ' (i \\ d £ d ' 7i - pK *\j d £ d ' T *- (5) 

The last member of this formula has a simple interpretation in terms of the 
artificial method of generating cyclic irrotational motion just explained. The 
first term has already been recognized as equal to twice the work done by 
the impulsive pressure pcf> applied to every part of the original boundary of 
the fluid. Again, pK X is the impulsive pressure applied, in the positive 
direction, to the infinitely thin massless membrane by which the place of the 
first barrier was supposed to be occupied ; so that the expression 



-»/j 



d< t> ^ 



denotes the work done by the impulsive forces applied to that membrane ; 
and so on. Hence (5) expresses the fact that the energy of the motion is equal 
to the work done by the whole system of impulsive forces by which we may 
suppose it generated. 

In applying (5) to the case where the fluid extends to infinity and is at 
rest there, we may replace the first term of the third member by 

-pjj(4>-C)QdS, (6) 

where the integration extends over the internal boundary only. The proof 



54-56] Kinetic Energy 57 

is the same as in Art 46. When the total flux across this boundary is zero, 
this reduces to 



P 



JK> ^ 



The minimum theorem of Lord Kelvin, given in Art. 45, may now be 
extended as follows: 

The irrotational motion of a liquid in a multiply-connected region has 
less kinetic energy than any other motion consistent with the same normal 
motion of the boundary and the same value of the total flux through each of 
the several independent channels of the region. 

The proof is left to the reader. 

Sources and Sinks. 

56. The analogy with the theories of Electrostatics, the Steady Flow 
of Heat, &c, may be carried further by means of the conception of sources 
and sinks. 

A ' simple source ' is a point from which fluid is imagined to flow out 
uniformly in all directions. If the total flux outwards across a small closed 
surface surrounding the point be m, then m is called the ' strength ' of the 
source. A negative source is called a 'sink.' The continued existence of 
a source or a sink would postulate of course a continual creation or annihila- 
tion of fluid at the point in question. 

The velocity-potential at any point P, due to a simple source, in a liquid 
at rest at infinity, is 

(f> = m/Awr, (1) 

where r denotes the distance of P from the source. For this gives a radial 
flow from the point, and if &S, = r 2 ^, be an element of a spherical surface 
having its centre at the source, we have 



-w 



d ^dS = m, 
or 



a constant, so that the equation of continuity is satisfied, and the flux outwards 
has the value appropriate to the strength of the source. 

A combination of two equal and opposite sources ± w', at a distance 8s 
apart, where, in the limit, Bs is taken to be infinitely small, and m' infinitely 
great, but so that the product m'Bs is finite and equal to yu, (say), is called 
a ' double source ' of strength /m, and the line Ss, considered as drawn in the 
direction from — m' to 4- m' , is called its axis. 

To find the velocity-potential at any point (a, y, z) due to a double source 



~"L[ 



58 Irrotational Motion [chap, hi 

fju situate at (#', y', z'), and having its axis in the direction (I, m, n), we remark 
that, / being any continuous function, 

/O' + IBs, y' + mBs, z' + nSs) -f{x\ y', z') 

ultimately. Hence, putting/ (a/, y' , z') — m' \kirr, where 
r ={{ x -oc'f + {y-y'? + (z-z'y}h 

wefind ^=H l i + m w +n i)^ (2) 

i<L + rn <L+ n <L)l (3) 

dx dy dzj r 1 

_ n cosS" ... 

-^T~^~' W 

where, in the latter form, ^ denotes the angle which the line r, considered as 
drawn from (V, y', z') to (%, y, z), makes with the axis (I, m, n). 

We might proceed, in a similar manner (see Art. 82), to build up sources 
of higher degrees of complexity, but the above is sufficient for our immediate 
purpose. 

Finally, we may imagine simple or double sources, instead of existing at 
isolated points, to be distributed continuously over lines, surfaces, or volumes. 

57. We can now prove that any continuous acyclic irrotational motion of 
a liquid mass may be regarded as due to a distribution of simple and double 
sources over the boundary. 

This depends on the theorem, proved in Art. 44, that if <£, <£' be any two 
single-valued functions which satisfy V 2 <£ = 0, V 2 0' = O throughout a given 
region, then 

IKt'^MK^ (5) 

where the integration extends over the whole boundary. In the present 
application, we take cj) to be the velocity-potential of the motion in question, 
and put <f> = 1/r, the reciprocal of the distance of any point of the fluid from 
a fixed point P. 

We will first suppose that P is in the space occupied by the fluid. Since 
</>' then becomes infinite at P, it is necessary to exclude this point from the 
region to which the formula (5) applies; this may be done by describing a 
small spherical surface about P as centre. If we now suppose S2 to refer to 
this surface, and &S to the original boundary, the formula gives 



56-58] Sources and Sinks 59 

At the surface 2 we have d/dn (1/r) = — 1/r 2 ; hence if we put 81=r 2 d&, 
and finally make r-*-0, the first integral on the left-hand becomes = — 4<7r</>p, 
where (f> P denotes the value of (f> at P, whilst the first integral on the right 
vanishes. Hence 

*--BjJfif«+eK®« « 

This gives the value of <£ at any point P of the fluid in terms of the values 
of <j> and d<f>/dn at the boundary. Comparing with the formulae (1) and (2) 
we see that the first term is the velocity-potential due to a surface distribution 
of simple sources, with a density — d<f>/dn per unit area, whilst the second term 
is the velocity-potential of a distribution of double sources, with axes normal 
to the surface, the density being <j>. It will appear from equation (10), below, 
that this is only one out of an infinite number of surface-distributions which 
will give the same value of throughout the interior. 

When the fluid extends to infinity in every direction and is at rest there, 
the surface-integrals in (7) may, on a certain understanding, be taken to refer 
to the internal boundary alone. To see this, we may take as external boundary 
an infinite sphere having the point P as centre. The corresponding part of 
the first integral in (7) vanishes, whilst that of the second is equal to G, the 
constant value to which, as we have seen in Art. 41, <£ tends at infinity. It 
is convenient, for facility of statement, to suppose (7 = 0; this is legitimate 
since we may always add an arbitrary constant to <£. 

When the point P is external to the surface, <f>' is finite throughout the 
original region, and the formula (5) gives at once 



*~cJ£&*+cKG)« < 8) 



where, again, in the case of a liquid extending to infinity, and at rest there, 
the terms due to the infinitely distant part of the boundary may be omitted. 

58. The distribution expressed by (7) can, further, be replaced by one of 
simple sources only, or of double sources only, over the boundary. 

Let </> be the velocity-potential of the fluid occupying a certain region, 
and let (/>' now denote the velocity- potential of any possible acyclic irrotational 
motion through the rest of infinite space, with the condition that </>, or <£', as 
the case may be, vanishes at infinity. Then, if the point P be internal to the 
first region, and therefore external to the second, we have 

°--ism^M*M)^\ 

where Bn, 8n' denote elements of the normal to dS, drawn inwards to the 



60 Irrotational Motion [chap, hi 

first and second regions respectively, so that d/dn' = — d/dn. By addition, we 
have 

♦.--sjj?e+&)«+sjj<*-*4©« ■■■» 

The function <f>' will be determined by the surface-values of <£' or d<j>/dn', 
which are as yet at our disposal. 

Let us in the first place make <f> = <j> at the surface. The tangential 
velocities on the two sides of the boundary are then continuous, but the normal 
velocities are discontinuous. To assist the ideas, we may imagine a liquid to 
fill infinite space, and to be divided into two portions by an infinitely thin 
vacuous sheet within which an impulsive pressure pcf> is applied, so as to 
generate the given motion from rest. The last term of (10) disappears, so that 



*>»*$&*%)"■ <"> 



that is, the motion (on either side) is that due to a surface-distribution of 
simple sources, of density 

Secondly, we may suppose that d<j>'/dn = d(f>/dn over the boundary. This 
gives continuous normal velocity, but discontinuous tangential velocity, over 
the original boundary. The motion may in this case be imagined to be 
generated by giving the prescribed normal velocity — d<\>jdn to every point 
of an infinitely thin membrane coincident in position with the boundary. The 
first term of (10) now vanishes, and we have 



fc-sjfa-^s©** < 12 > 



shewing that the motion on either side may be conceived as due to a surface- 
distribution of double sources, with density 

It may be shewn that the above representations of (f> in terms of simple 
sources alone, or of double sources alone, are unique; whereas the representa- 
tion of Art. 57 is indeterminate f. 

It is obvious that cyclic irrotational motion of a liquid cannot be reproduced by any 
arrangement of simple sources. It is easily seen, however, that it may be represented by 
a certain distribution of double sources over the boundary, together with a uniform distri- 
bution of double sources over each of the barriers necessary to render the region occupied 
by the fluid simply-connected. In fact, with the same notation as in Art. 53, we find 

* This investigation was first given by Green, from the point of view of Electrostatics, I.e. 
ante p. 46. 

f Cf. Larmor, "On the Mathematical Expression of the Principle of Huyghens," Proc. Lond. 
Math. Soc. (2) i. 1 (1903) [Math, and Phys. Papers, Cambridge, 1929, ii. 240]. 



58] Surface-Distributions 61 

where <p is the single-valued velocity-potential which obtains in the modified region, and 
<f>' is the velocity-potential of the acyclic motion which is generated in the external space 
when the proper normal velocity —dcfy/dn is given to each element 8S of a membrane 
coincident in position with the original boundary. 

Another mode of representing the irrotational motion of a liquid, whether 
cyclic or not, will present itself in the chapter on Vortex Motion. 

We here close this account of the theory of irrotational motion. The 
mathematical reader will doubtless have noticed the absence of some im- 
portant links in the chain of our propositions. For example, apart from 
physical considerations, no proof has been offered that a function <f> exists 
which satisfies the conditions of Art. 36 throughout any given simply- 
connected region, and has arbitrarily prescribed values over the boundary. 
The formal proof of 'existence-theorems' of this kind is not attempted in 
the present treatise. For a review of the literature of this part of the 
subject the reader may consult the authors cited below*. 

* H. Burkhardt and W. F. Meyer, "Potentialtheorie," and A. Sommerfeld, "Randwerth- 
aufgaben in der Theorie d. part. Diff.-Gleichungen," Encyc. d. math. Wiss. ii. (1900). 



CHAPTEE IV 

MOTION OF A LIQUID IN TWO DIMENSIONS 

59. If the velocities u, v be functions of x, y only, while w is zero, the 
motion takes place in a series of planes parallel to xy, and is the same in 
each of these planes. The investigation of the motion of a liquid under these 
circumstances is characterized by certain analytical peculiarities; and the 
solutions of several problems of great interest are readily obtained. 

Since the whole motion is known when we know that in the plane z = 0, 
we may confine our attention to that plane. When we speak of points and 
lines drawn in it, we shall understand them to represent respectively the 
straight lines parallel to the axis of z, and the cylindrical surfaces having 
their generating lines parallel to the axis of z> of which they are the traces. 

By the flux across any curve we shall understand the volume of fluid 
which in unit time crosses that portion of the cylindrical surface, having the' 
curve as base, which is included between the planes z = 0, z — 1. 

Let i, P be any two points in the plane xy. The flux across any two 
lines joining AP is the same, provided they can be reconciled without passing 
out of the region occupied by the moving liquid; for otherwise the space 
included between these two lines would be gaining or losing matter. Hence 
if A be fixed, and P variable, the flux across any line AP is a function of the 
position of P. Let yfr be this function; more precisely, let yjr denote the flux 
across A P from right to left, as regards an observer placed on the curve, and 
looking along it from A in the direction of P. Analytically, if I, m be the 
direction-cosines of the normal (drawn to the left) to any element 8s of the 
curve, we have 

yfr= I (lu + mv)ds (1) 

If the region occupied by the liquid be aperiphractic (see p. 40), i/r is neces- 
sarily a single-valued function, but in periphractic regions the value of -v/r 
may depend on the nature of the path A P. For spaces of two dimensions, 
however, periphraxy and multiple-connectivity become the same thing, so that 
the properties of i/r, when it is a many-valued function, in relation to the 
nature of the region occupied by the moving liquid, may be inferred from 
Art. 50, where we have discussed the same question with regard to </>. The 
cyclic constants of yjr, when the region is periphractic, are the values of the 
flux across the closed curves forming the several parts of the internal 
boundary. 



59-eo] Stream- Function 63 

A change, say from A to B, of the point from which yjr is reckoned has 
merely the effect of adding a constant, viz. the flux across a line BA, to the 
value of yfr; so that we may, if we please, regard ijr as indeterminate to the 
extent of an additive constant. 

If P move about in such a manner that the value of yjr does not alter, it 
will trace out a curve such that no fluid anywhere crosses it, i.e. a stream-line. 
Hence the curves yjr = const, are the stream-lines, and -\jr is called the 'stream- 
function.' 

If P receive an infinitesimal displacement PQ (= By) parallel to y, the 
increment of yfr is the flux across PQ from right to left, i.e. 8yjr = — u. PQ, or 

«-- 1 ^ 

Again, displacing P parallel to x, we find in the same way 

-3 < 3 > 

The existence of a function \jr related to u and v in this manner might also 
have been inferred from the form which the equation of continuity takes in 
this case, viz. 

du dv n " 

3 -* + ar 0> "; ; (4) 

which is the analytical condition that udy — vdx should be an exact 
differential *. 

The foregoing considerations apply whether the motion be rotational or 
irrotational. The formulae for the components of vorticity, given in Art. 30, 
become 

*-* <-* *-£'■-■?£ < 5 > 

so that in irrotational motion we have 

dx 2 + dy 2 w 

60. In what follows we confine ourselves to the case of irrotational 
motion, which is, as we have already seen, characterized by the existence, in 
addition, of a velocity-potential <£, connected with u, v by the relations 

d<j> dcf> 

U = ~dx> V = ~ d y> W 

and, since we are considering the motion of incompressible fluids only, 
satisfying the equation of continuity 

dx 2 ' dy' 



+H-° ( 2 ) 



* The function \j/ was introduced in this way by Lagrange, Noav. mem. de VAcad. de Berlin, 
1781 [Oeuvres, iv. 720]. The kinematical interpretation is due to Eankine, "On Plane Water- 
Lines in Two Dimensions," Phil. Trails. 1864 [Miscellaneous Scientific Papers, London, 1881, 
p.,495]. 



64 Motion of a Liquid in Two Dimensions [chap, iv 

The theory of the function (f>, and the relation between its properties and 
the nature of the two-dimensional space through which the irrotational 
motion holds, may be readily inferred from the corresponding theorems in 
three dimensions proved in the last chapter. The alterations, whether of 
enunciation or of proof, which are requisite to adapt these to the case of two 
dimensions are for the most part purely verbal. 

For instance, we have the theorem that the mean value of <f> over the 
circumference of a circle is equal to its value at the centre, provided the circle 
can be contracted to a point, remaining always within the region occupied by 
the fluid. 

Again, if this region extends to infinity, being bounded internally by one 
or more closed curves, and if the velocities tend to a zero limit at infinity, the 
value of <f> tends there to a constant limit, provided the total flux across the 
internal boundaries is zero. This latter proviso is now essential. 

The fundamental solution of the equation (2) has the form <£ = G log r, 
where r denotes distance from a fixed point. This is the case of a two-dimen- 
sional source, for if we write 

* — £logr (3) 

the flux outwards across a circle surrounding the point is 

-^. 27rr = ra (4) 

The constant m accordingly measures the output, or ' strength ', of the source. 
We get essentially the same result if we imagine point sources of the type 
explained in Art. 56 to be distributed with uniform line-density m along its 
axis of z. The velocity in that case will be in the direction of r, and equal to 
ra/27rr, consistently with (3). We have here the conception of a 'line-source' 
(in three dimensions). 

For a double source, or 'doublet', as it is sometimes called, we have the 
formula 

*-ll^r) (5) 

where the symbol d/ds indicates a space-differentiation in the direction of the 
axis of the source. If ^ be the angle which direction of r increasing makes 
with this axis, we have Br = — Bs cos ^, and therefore 

* = £^ W 

Again we might establish a system of formulae analogous to those of Art. 58. 
In particular, corresponding to Art. 58 (12), we have 



fc— i/(* -*') J; ( lo s *•><**• < 7 > 



60-60 a] Electrical Analogies 65 

giving the value of (/> in any region in terms of a distribution of double sources 
over the boundary. This will apply to the case of a fluid unlimited externally, 
provided the velocities tend to zero at infinity, and that the total flux outwards 
is zero. As in Art. 58 the function </>' refers to the space within the inner 
boundary, and is subject to the condition that d<f>'/dn = d<j>/dn at this boundary. 
A deduction from this formula will be given presently (Art. 72 a). 

60 a. The foregoing kinematical relations have exact analogies in the theory 
of electric conduction. In the case of a uniform plane sheet we have 

. dv dv m 

^-~ 5P ag = -W () 

with ¥+¥ = <>> W 

dec oy 

where (/, g) is the current density, V is the electric potential, and a is the 
specific resistance of the material. If we write 

u = af, v = ag t <f>=V, (3) 

these become identical with the hydrodynamical relations. This has suggested 
a practical method of solution of two-dimensional hydrokinetic problems. The 
current sheet may consist of a thin layer of feebly conducting fluid (H 2 S0 4 ) 
contained in a rectangular tank, two opposite walls of which are metallic and 
maintained at a constant difference of potential whilst the remaining walls 
(and the bottom) are insulators. The equipotential lines, to which the current 
lines are orthogonal, are easily traced electrically, and in this way practical 
solutions can be obtained of problems of flow of a stream past an obstacle 
(represented by a non-conducting disk in the electrical experiment) which are 
not easily treated by analysis*. 

Again, instead of (3) we may put 

u = — ag, v = crf, -\jr = — V. (4) 

The hydrodynamical relations are satisfied, but the stream- lines are now 
represented by the lines of equal electric potential, and can therefore be found 
directly. An obstacle has now to be represented by a disk whose conductivity 
so greatly exceeds that of the surrounding stratum that it may be regarded 
as practically perfect. This analogy has the further advantage that circulation 
can also be represented. For if (I, m) be the direction of the outward normal 
to the contour of the obstacle, the circulation is 

j(lv — mu)ds= a j(lf+ mg)ds, (5) 

* For experimental details reference should be made to E. F. Relf , Phil. Mag. (6) xlviii. (1924). 
As a test of the method the diagram on p. 86 infra was reproduced with remarkable accuracy. 
The circulation round a lamina was also determined and compared with theory. 



66 Motion of a Liquid in Two Dimensions [chap, iv 

and is therefore proportional to the total current outwards in the electric 
analogy. For this purpose the disk is connected with one terminal of a suitable 
battery, the other terminal being connected with one of the conducting walls 
of the tank. 

61. The kinetic energy I 1 of a portion of fluid bounded by a cylindrical 
surface whose generating lines are parallel to the axis of z, and by two 
planes perpendicular to the axis of z at unit distance apart, is given by the 
formula 

»-'/Ji©" + (gn**~' \*i* « 

where the surface-integral is taken over the portion of the plane xy cut off 
by the cylindrical surface, and the line-integral round the boundary of this 
portion. Since d<fi/dn =■■ — df/ds, the formula (1) may be written 

2T = pf<t>d+, (2) 

the integration being carried in the positive direction round the boundary. 

If we attempt by a process similar to that of Art. 46 to calculate the energy in the case 
where the region extends to infinity, we find that its value is infinite, except when the total 
flux outwards (M) is zero. For if we introduce a circle of great radius r as the external 
boundary of the portion of the plane xy considered, we find that the corresponding part 
of the integral on the right-hand side of (1) increases indefinitely with r. The only excep- 
tion is when M=0, in which case we may suppose the line-integral in (1) to extend over 
the internal boundary only. 

If the cylindrical part of the boundary consist of two or more separate 
portions one of which embraces all the rest, the enclosed region is multiply- 
connected, and the equation (1) needs a correction, which may be applied 
exactly as in Art. 55. 



* Conformal Transf of motions. 
62. The functions <f> and f are connected by the relations 

d± = Hi ^ = _?^ (if-* 

dx dy ' dy dx ' '" ** 

These conditions are fulfilled by equating <f> + if, where i stands as usual 
for \/( — 1 ), to any ordinary algebraic or transcendental function of x + iy, say 

</> + *> =/(* + «» (2) 

For then g- (0 4- if) = if (x + iy) = i — (<f> + if), (3) 

whence, equating separately the real and the imaginary parts, we see that the 
equations (1) are satisfied. 



60a-62] Complex Variable 67 

Hence any assumption of the form (2) gives a possible case of irrotational 
motion. The curves (j> = const, are the curves of equal velocity-potential, and 
the curves yjr = const, are the stream -lines. Since, by (1), 

d(j> d-yjr dcf) dyjr _ 

dx dx dy dy 

we see that these two systems of curves cut one another at right angles, as 
already proved. Since the relations (1) are unaltered when we write — -v/r for 
(f>, and cf> for yfr, we may, if we choose, look upon the curves ty = const, as the 
equipotential curves, and the curves <£ = const, as the stream-lines; so that 
every assumption of the kind indicated gives us two possible cases of 
irrotational motion. 

For shortness, we shall through the rest of this chapter follow the usual 
notation of the Theory of Functions, and write 

z = x + iy, (4) 

W = $ 4- iyfr (5) 

From a modern point of view, the fundamental property of a function 
of a complex variable is that it has a definite differential coefficient with 
respect to that variable*. If cf), ijr denote any functions whatever of x and y, 
then corresponding to every value of x + iy there must be one or more 
definite values of <£ + iyjr; but the ratio of the differential of this function 
to that of x + iy, viz. 

8+ + JS+ or @ + *to) fe+ C| + ^) Sy 
&x+ i8y ' Sso + i8y 

depends in general on the ratio hx : hy. The condition that it should be the 
same for all values of the latter ratio is 

d A +i w =i m +i m (6) 

dy dy \dx dx J ' 

which is equivalent to (1) above. This property was adopted by Riemann 
as the definition of a function of the complex variable x 4- iy; viz. such 
a function must have, for every assigned value of the variable, not only a 
definite value or system of values, but also for each of these values a definite 
differential coefficient. The advantage of this definition is that it is quite 
independent of the existence of an analytical expression for the function. 

If the complex quantities z and w be represented geometrically after 
the manner of Argand and Gauss, the differential coefficient dwjdz may be 
interpreted as the operator which transforms an infinitesimal vector Bz into 
the corresponding vector Sw. It follows then, from the above property, that 
corresponding figures in the planes of z and w are similar in their infinitely 
small parts. 

* See, for example, Forsyth, Theory of Functions, 3rd ed., Cambridge, 1918, cc. i., ii. 



68 Motion of a Liquid in Two Dimensions [chap, iv 

For instance, in the plane of w the straight lines <j> = const., yjr = const., 
where the constants have assigned to them a series of values in arithmetical 
progression, the common difference being infinitesimal and the same in each 
case, form two systems of straight lines at right angles, dividing the plane 
into infinitely small squares. Hence in the plane xy the corresponding curves 
<f> = const., yjr = const., the values of the constants being assigned as before, 
cut one another at right angles (as has already been proved otherwise) and 
divide the plane into infinitely small squares. 

Conversely, if 0, yjs be any two functions of x, y such that the curves = me, \jr = ne, 
where e is infinitesimal, and m, n are any integers, divide the plane xy into elementary 
squares, it is evident geometrically that 

dx _ dy dx _ _dy 

If we take the upper signs, these are the conditions that x + iy should be a function of 
(j> + iyfr. The case of the lower signs is reduced to this by reversing the sign of yj/. Hence 
the equation (2) contains the complete solution of the problem of conformal representation 
of one plane on another*. 

The similarity of corresponding infinitely small portions of the planes w 
and z breaks down at points where the differential coefficient dw/dz is zero 
or infinite. Since 

d ™ = f + i d -±, (7)* 

dz ox Ox 

the corresponding value of the velocity, in the hydrodynamical application, is 
zero or infinite. 

In all physical applications, w must be a singl^yaljied, or at mos^L-cyclic 
function of z in the sense of Art. 50, throughout the region with which we 
are concerned. Hence in the case of a 'multiform' function, this region must 
be confined to a single sheet of the corresponding Riemann's surface, and 
'branch-points' therefore must not occur in its interior. 

63. We can now proceed to some applications of the foregoing method. 
First let us assume w = A z n , 

A being real. Introducing polar co-ordinates, r, 0, we have 

(j> = Ar n cos nO, 



y\r = Ar n sin nO. [ 

The following cases may be noticed. 

1°. If n « 1, the stream-lines are a system of straight lines parallel to x, 
and the equipotential curves are a similar system parallel to y. In this case 
any corresponding figures in the planes of w and z are similar, whether they 
be finite or infinitesimal. 

* Lagrange, " Sur la construction des cartes geographiques," Nouv. mem. de VAcad. de Berlin, 
1779 [Oeuvres, iv. 636]. For the further history of the problem, see Forsyth, Theory of Functions, 
c. xix. 



62-64] Examples 69 

2°. If n = 2, the curves <f> = const, are a system of rectangular hyperbolas 
having the axes of co-ordinates as their principal axes, and the curves 
yfr = const, are a similar system, having the co-ordinate axes as asymptotes. 
The lines 6 = 0, 6 = ^ir are parts of the same stream-line yfr = 0, so that we 
may take the positive parts of the axes of x, y as fixed boundaries, and thus 
obtain the case of a fluid in motion in the angle between two perpendicular 
walls. 

3°. If n = — 1, we get two systems of circles touching the axes of 
co-ordinates at the origin. Since now <£ = A/r . cos 6, the velocity at the 
origin is infinite; we must therefore suppose the region to which our formulae 
apply to be limited internally by a closed curve. 

4°. If n = — 2, each system of curves is composed of a double system of 
lemniscates. The axes of the system <f> = const, coincide with x or y; those 
of the system -*|r = const, bisect the angles between these axes. 

5°. By properly choosing the value of n we get a case of irrotational 
motion in which the boundary is composed of two rigid walls inclined at any 
angle a. The equation of the stream-lines being 

r 11 sin nO = const., (2) 

we see that the lines 6 = 0, 6 = irjn are parts of the same stream-line. 
Hence if we put n = irja, we obtain the required solution in the form 

(/> = Ar a cos — > yfr = Ar a sin — {?) 

The component velocities along and perpendicular to r are 

— A-r a cos~, and A~r a sin— > \V 

and are therefore zero, finite, or infinite at the origin, according as a is less 
than, equal to, or greater than ir. 

64. We take some examples of cyclic functions. 

1°. The assumption w = — /j,\ogz, (1) 

where /j, is real, gives <f> = — /jl log r, ^ = — fi6 (2) 

The velocity at a distance r from the origin is fi/r; this point must therefore 
be isolated by drawing a closed curve round it. 

If we take the radii 6 = const, as the stream-lines we get the case of 
a (two-dimensional) source, of strength lir^ at the origin. (See Art. 60.) 

If the circles r = const, be taken as stream-lines we have the case of 
Art. 27; the motion is now cyclic, the circulation in any circuit embracing 
the origin being 2irfjL. 



70 



Motion of a Liquid in Two Dimensions [chap, iv 



2°. Let us take 



W = — fJb log 



z — a 
z + a 



(3) 



If we denote by r 1} r 2 the radii drawn to any point in the plane xy from 
the points (+ a, 0), and by 9 ly 6 2 the angles which these radii make with the 
positive direction of the axis of x, we have 

z — a = ri e^ 1 , z + a = r 2 e i02 , 

whence <$> — — f^logr^^, ^ = — ^(^1 — ^2) (4) 

The curves $ = const., ^ = const, form two orthogonal systems of 'coaxal' 
circles. 




Either of these systems may be taken as the equipotential curves, and 
the other system will then form the stream-lines. In either case the velocity 
at the points (± a, 0) will be infinite. If these points be accordingly isolated 
by drawing closed curves round them, the rest of the plane xy becomes 
a triply-connected region. 

If the circles X — 6 2 = const, be taken as the stream-lines we have the case 
of a source and a sink, of equal intensities, situate at the points (+ a, 0). If a is 
diminished indefinitely, whilst /xa remains finite, we reproduce the assumption 
of Art. 60 (5), which corresponds to the case of a double line-source at the 
origin. The lines of motion are shewn (in part) on p. 76. 

If, on the other hand, we take the circles ?\/r 2 = const, as the stream-lines 
we get a case of cyclic motion, viz. the circulation in any circuit embracing 



64] A How of Sources 71 

the first (only) of the above points is 27r/x, that in a circuit embracing the 
second is - 27t/a; whilst that in a circuit embracing both is zero. This example 
will have additional interest- for us when in Chapter VII. we come to treat of 
' Rectilinear Vortices.' 

3°. By a simple combination of sources we can represent the flow past a circular barrier 
due to a source at a given external point P. 

Let Q be the inverse point of P with respect to the circle, and imagine equal sources p at 



— X 




P and Q, and a sink - \i at the centre 0. Then, referring to (2) above, the value of ^ at a 
point R on the circumference is 

^=-lL{RPX+RQX-R0X)=- i L(RPX+0RQ)=- l x(RPX+RP0)=-7rn, 
a constant over the circle*. 

4°. The potential- and stream-functions due to a row of equal and equidistant sources 
at the points (0, 0), (0, ±a), (0, ±2a), ... are given by the formula 

w oc log z + log (z - ia) + log (z - ia) + log (z - 2ia) + log (z + 2ia) + . . ., (5) 

or, say, w = (71ogsinh^- , (6) 

where C is real. This makes 



, 1 m 1/ i^ 71 *- 27 2?r #\ / m. i ftan(iry/a) 1 

d> = - C log -cosh cos— & , \^=<7tan-Mr — u , , \ \ 

^2 & 2\ a a J r (tanh (nx a)) 



•(7) 



in agreement with a result given by Maxwell t. The formulae apply also to the case of a 
source midway between two fixed boundaries y= ±^a. 

The case of a row of double sources having their axes parallel to x is obtained by- 
differentiating (6) with respect to z. Omitting a factor we have 

w = Ccoth — , (8) 

Csinh(27rx/a) _ Csm(27ry/a) 
^ — cosh (27rx/a)- cos (fliry/a) 3 ^ cosh (2nx/a) - cos (2rry/a) ' * ^ ' 

Superposing a uniform motion parallel to x negative, we have 

w = z + C coth — , (10) 

Cainh. (2scxJa) . C sin (2iry Id) ,,,. 

qji (b=x-\- - - - vs = v ^ oil ni^ 

r cosh (ZTrxja) - cos {2iry\a) ' y ^ cosh (2irx/a) — cos (Siry/a) ' *" v ; 

The stream-line ^ = now consists in part of the line y = 0, and in part of an oval curve 
whose semi-diameters parallel to x and y are given by the equations 

8mn 2!^ = I^ y tan- y = a (12) 

* Kirchhoff, Pogg. Ann., lxiv. (1845) [Ges. Abh. 1]. 
f Electricity and Magnetism, Art. 203. 



72 Motion of a Liquid in Two Dimensions [chap, iv 

If we put <7=7r6 2 /a, (13) 

where b is small compared with a* these semi-diameters are each equal to b, approxi- 
mately. We thus obtain the potential- and stream-functions for a liquid flowing through a 
grating of parallel cylindrical bars of small circular section. The second of equations (11) 
becomes in fact, for small values of x, y, 



M 1 -;^?) (14) 



65. If w be a function of z, it follows at once from the definition of Art. 62 
that z is a function of w. The latter form of assumption is sometimes more 
convenient analytically than the former. 

The relations (1) of Art. 62 are then replaced by 

dx = dy_ fa dy £ 

d4> fyr' df a<£ K } 

A , dw dd> dyjr 

Also since -y- - = K - + % ^- = — u 4- iv, 

dz doc dx 



i dz 1 

we have — — = - 

dw u — iv 



q\q qJ 



where q is the resultant velocity at (x, y). Hence if we write 

<>-£■ (2) 

and imagine the properties of the function f to be exhibited graphically in 
the manner already explained, the vector drawn from the origin to any point 
in the plane of f will agree in direction with, and be in magnitude the 
reciprocal of, the velocity at the corresponding point of the plane of z. 

Again, since 11 q is the modulus of dz/dw, i.e. of dx/d<f> + idy/d<j>, we have 

K-f)'+© ! <*> 

which may, by (1), be put into the equivalent forms 

(4) 

The last formula, viz. -, S |M , (5) 

expresses the fact that corresponding elementary areas in the planes of z and 
w are in the ratio of the square of the modulus of dz/dw to unity. 

* The approximately circular form holds however for a considerable range of values of C. 
Thus if we put C=±a, we find from (12) 

x/a =-254, y/a = -250. 

The two diameters are very nearly equal, although the breadth of the oval is half the interval 
between the stream-lines y= =*=£a. 



64-66] Inverse Methods 73 

66. The following examples of this procedure are important. 
1°. Assume z — ccoshw, (1) 

or x — c cosh <f> cos -\jr,) .^\ 

y = c sinh <f> sin yfr. J 

The curves <£> = const, are the ellipses 

^ i V __ -I (o\ 

c 2 cosh 2 (/> c 2 sinh 2 </> ' * v ; 

and the curves yjr = const, are the hyperbolas 

_^ l^_ = l (4) 

C 2 cos 2 yjr c 2 sin 2 yfr ' 

these conies having the common foci (+ c, 0). The two systems of curves are 
shewn below. 




Since at the foci we have = 0,-^ = rnr, n being some integer, we see by 
(2) of the preceding Art. that the velocity there is infinite. If the hyperbolas 
be taken as the stream-lines, the portions of the axis of at which lie outside 
the points (± c, 0) may be taken as rigid boundaries. We obtain in this 
manner the case of a liquid flowing from one side to the other of a thin plane 
partition, through an aperture of breadth 2c. 



74 Motion of a Liquid in Two Dimensions [chap, iv 

If the ellipses be taken as the stream-lines we get the case of a liquid 
circulating round an elliptic cylinder, or, as an extreme case, round a lamina 
whose section is the line joining the foci (+ c, 0). 

At an infinite distance from the origin <j> is infinite, of the order logr, 
where r is the radius vector; and the velocity is infinitely small of the order 1/r. 

2°. Let z = w + e w , (5) 

or x = (/> + e* cos -\jr, y = i/r + e^ sin yjr (6) 

The stream-line ty = coincides with the axis of x. Again, the portion of the 
line y = tt between x = — oo and x = — 1, considered as a line bent back on 





itself, forms the stream -line ^r = ir; viz. as <j> decreases from -+- oo through 
to — oo , x increases from — oo to — 1 and then decreases to — oo again. 
Similarly for the stream-line -vjr = — tt. 

Since f = — dzjdw = — 1 — efi cos ty — id> sin \^, 

it appears that for large negative values of (f> the velocity is in the direction 
of ^-negative, and equal to unity, whilst for large positive values it is zero. 

The above formulae therefore express the motion of a liquid flowing into 
a canal bounded by two thin parallel walls from an open space. At the ends 
of the walls we have </> = 0, \/r => ± tt, and therefore f=0, i.e. the velocity is 



66-67] General Formulae 75 

infinite. The forms of the stream-lines, drawn, as in all similar cases in this 
chapter, for equidistant values of ^, are shewn in the figure on p. 74*. 

If the walls instead of being parallel make angles ±/3 with the line of symmetry, the 
appropriate formula is 

z=l—0: (l-e-»«')+e( 1 - n > w } (7) 

n 

where »=£/*-. The stream-lines \^=±tt follow the course of the walls t. This agrees 
with (5) when n tends to the limit 0, whilst if n = \ we have virtually the case shewn on 
p. 73. 

If we change the sign of w in (5) the direction of flow is reversed. If we further super- 
pose a uniform stream in the negative direction of %, by writing w-z for w, we obtain { 

w = e z ~ w , or z = w + \ogiv (8) 

The velocity between the walls at a great distance to the left is now annulled, and we 
have an idealized representation of a Pitot tube (Art. 24). The stream-lines can be plotted 
from the formulae 

.z = (£+*log((£ 2 + xP), .y = ^ + tan" 1 (W) (9) 

67. It is known that a function f(z) which is finite, continuous, and 
single-valued, and has its first derivative finite, at all points of the space 
included between two concentric circles about the origin, can be expanded 
in the form 

f(z) = A + A 1 z+A 2 z*+...+B 1 z- 1 + B 2 z- 2 + (1) 

If the above conditions be satisfied at all points within a circle having the 
origin as centre, we retain only the ascending series ; if at all points without 
such a circle, the descending series, with the addition of the constant A , is 
sufficient. If the conditions be fulfilled for all points of the plane xy without 
exception,/ (z) can be no other than a constant A . 

Putting /(y)=<£+^, introducing polar co-ordinates, and writing the 
complex constants A nt B n in the forms P n +iQn, R n + i8n, respectively, 
we obtain- 
ed = P + 2T r n (P n cos nO - Q n sin n&) + 2? r~ n (R n cos nd + £ n sin nd\] 



yfr= Q + 2f r n (Q n cos n6 - P n sin nd) + 25° r~ r > (S n cos nO -R n sin n6).\ ' ' 

These formulae are convenient in treating problems where we have the 
value of cj), or of d(f>/dn, given over concentric circular boundaries. This 
value may be expanded for each boundary in a series of sines and cosines of 
multiples of 6, by Fourier's theorem. The series thus found must be equi- 
valent to those obtained from (2); whence, equating separately coefficients 
of sin n6 and cos n6 y we obtain equations to determine P n , Q n , R n , S n . 

* This example was given by Helmholtz, Berl. Monatsber. April 23, 1868 [Phil. Mag. Nov. 
1868 ; Wiss. Abh. i. 154]. 

t K. A. Harris, "On Two-Dimensional Fluid Motion through Spouts composed of two Plane 
Walls," Ann. of Math. (2), ii. (1901). A diagram is given for the case of j3 = $Tr. 

X Kayleigh, Proc. Roy. Soc. A, xci. 503 (1915) [Papers, vi. 329], where a few of the stream- 
lines are traced. 



76 



Motion of a Liquid in Two Dimensions [chap, iv 



68. As a simple example let us take the case of an infinitely long circular 
cylinder of radius a moving with velocity U perpendicular to its length, in an 
infinite mass of liquid which is at rest at infinity. 

Let the origin be taken in the axis of the c}dinder, and the axes of x, y 
in a plane perpendicular to its length. Further let the axis of x be in the 
direction of the velocity U. The motion, supposed originated from rest, will 
necessarily be irrotational, and </> will be single-valued. Also, since fd(j>/dn.ds, 
taken round the section of the cylinder, is zero, yjr is also single-valued 
(Art. 59), so that the formulae (2) apply. Moreover, since dcf)/dn is given at 
every point of the internal boundary of the fluid, viz. 



— ■=— = U cos 6. for r — a. 
dr 



(3) 




and since the fluid is at rest at infinity, the problem is determinate, by 
Art. 41. These conditions give P n = 0, Q n = 0, and 

Ucos6 = ST nor 71 - 1 (R n cos nO + S n sin nO), 

which can only be satisfied by making Rx = Ua z , and all the other coefficients 
zero. The complete solution is therefore 

Ua 2 



a Ua2 a 

<b — — — cos 6, 

r 



t- 



sin#. 



■{*) 



The stream-lines yjr = const, are circles, as shewn above. Comparing with 
Art. 60 (6) we see that the effect is that of a double source at the origin. 



68] Motion of a Cylinder 77 

The kinetic energy of the liquid is given by the formula (2) of Art. 61, viz. 
2T = p Udyjr = P U 2 a 2 rcos 2 ed6 = M'U 2 } (5) 

if M', = ira 2 p, be the mass of fluid displaced by unit length of the cylinder. 
This result shews that the whole effect of the presence of the fluid may be 
represented by an addition M ' to the inertia per unit length of the cylinder. 
Thus, in the case of rectilinear motion, if we have an extraneous force X per 
unit length acting on the cylinder, the equation of energy gives 

or (M+M') d ^=X, (6) 

where M represents the mass of the cylinder itself. 
Writing this in the form 

at at 

we learn that the pressure of the fluid is equivalent to a force —M'dU/dt 
per unit length in the direction of motion. This vanishes when U is constant. 

The above result can be verified by direct calculation. By Art. 20 (7), (8) the pressure 
is given by the formula 

«.a \-if + F(o, ( 7) 

p vt 

provided q denotes the velocity of the fluid relative to the axis of the moving cylinder. 
The term due to the extraneous forces (if any) acting on the fluid has been omitted ; the 
effect of these would be given by the rules of Hydrostatics. We have, for r = a, 

^t = a ^GOS0 f ? a = 4^ 2 sin 2 0, (8) 

whence p=p (a — cob 0-2U 2 sin 2 0+F(t)\ (9) 

The resultant force on unit length of the cylinder is evidently parallel to the initial line 
0=0; to find its amount we multiply by —add. cos and integrate with respect to 6 
between the limits and 2n. The result is -M'dU/dt, as before. 

If in the above example we impress on the fluid and the cylinder a 
velocity — U we have the case of a current flowing with the general velocity 
U past a fixed circular cylinder. Adding to cj> and ^r the terms Ur cos 6 and 
Ur sin 0, respectively, we get 

$= U L.+ -)cos<9, ^= U(r- -) sin 6> (10) 

The stream-lines are shewn on the next page. 

If no extraneous forces act, and if U be constant, the resultant force on 
the cylinder is zero. Cf. Art. 92. 



78 



Motion of a Liquid in Two Dimensions [chap, iv 




69. To render the formula (1) of Art. 67 capable of representing any 
case of continuous irrotational motion in the space between two concentric 
circles, we must add to the right-hand side the term 

^-log* (1) 

If A = P + iQ, the corresponding terms in <fi, ty are 

P\ogr-Qd, P0 + Qlogr, (2) 

respectively. The meaning of these terms is evident ; thus 2irP, the cyclic 
constant of -v/r, is the flux across the inner (or outer) circle ; and 2irQ, the 
cyclic constant of </>, is the circulation in any circuit embracing the origin. 

For example, returning to the problem of the last Art., let us suppose that 
in addition to the motion produced by the cylinder we have an independent 
circulation round it, the cyclic constant being k. The boundary-condition is 
then satisfied by 



<f> = U — cos 6 

T r 



6. 



\ir 



(3) 



The effect of the cyclic motion, superposed on that due to the cylinder, 
will be to augment the velocity on one side, and to diminish (and, it may be, 
to reverse) it on the other. Hence when the cylinder moves in a straight 
line with constant velocity, there will be a diminished pressure on one side, 
and an increased pressure on the other, so that a constraining force must be 
applied at right angles to the direction of motion. 



69] 



Cylinder with Circulation 



79 



The figure shews the lines of flow. At a distance from the origin they approximate to 
the form of concentric circles, the disturbance due to the cylinder becoming small in com- 
parison with the cyclic motion. When, as in the case represented, {7>K/2xra, there is a 
point of zero velocity in the fluid. The stream -line system has the same configuration in 
all cases, the only effect of a change in the value of U being to alter the radius of the 
cylinder on the scale of the diagram. 




When the problem is reduced to one of steady motion we have in place of (3) 

4=u(r + ^coa0-£-6, (4) 

whence - = const. — \ a' 2 

P 

= const. -U 2U sin + cp— ) , (5) 

for r = a. The resultant pressure on the cylinder is therefore 

r2tr 

— / p sin 6 a d6 — + <p U, (6) 

at right angles to the general direction of the stream. This result is independent of the 
radius of the cylinder. It will be shewn later that it holds for any form of section*. 

To calculate the effect of the fluid pressures on the cylinder when moving in any 
manner we may conveniently adopt moving axes, the origin being taken at the centre^ 
and the axis of x in the direction of the velocity U. If x De the angle which this makes 
with a fixed direction, the equation (6) of Art. 20 gives 

p~ dt ^ q dt dd ' { ' 

* This important theorem is due to Kutta and Joukowski; see Kutta, Sitzb. d. h. bayr. Akad. 
d. Wiss. 1910. Proofs are given later (Arts. 72 b, 372). 



80 Motion of a Liquid in Two Dimensions [chap, iv 

where q now denotes fluid velocity relative to the origin, to be calculated from the relative 
velocity-potential (f>+ Ur cos 6, <p being given by (3). We find, for r=a, 

f-f-'-i^'-'+^-^'+sS < 8 > 

The resultant pressures parallel to x and y are therefore 

/27T J ' TJ /■ 2JT ■% 
pcos0ad6=-M'-j- t , - psm3ad0 = K P U-M'U^, (9) 

where M' = irpa 2 as before. 

Hence, if P, Q denote the components of the extraneous forces, if any, acting on the 
cylinder in the directions of the tangent and the normal to the path, respectively, the 
equations of motion of the cylinder are 

(Jf+JfO-5-P, ) 

, \ (10) 

iM+M^uSjt-KpU+Q.) 

If there be no extraneous forces, U is constant, and writing dx/dt = U/R, where R is 
the radius of curvature of the path, we find 

R=(M + M')U/k P (11) 

The path is therefore a circle, described in the direction of the cyclic motion* 

If £, t) be the Cartesian co-ordinates of a point on the axis of the cylinder relative to 
fixed axes, the equations (10) are equivalent to 

{M+M'y^-Kptj+XA 

(M+Myr}= <pi+r,j K } 

where X, Y are the components of the extraneous forces. To find the effect of a constant 
force, we may put 

X=(M+M')g' i T=0 (13) 

The solution then is £ = a + ccos (nt + €) t \ 

q > (14) 

17= /8 +Z-t+c sin (nt+e\ 

where a, ft, c, e are arbitrary constants, and 

n = <p/(M+M') (15) 

This shews that the path is a trochoid, described with a mean velocity g'/n perpendicular 
to x t. It is remarkable that the cylinder has on the whole no progressive motion in the 
direction of the extraneous force. In the particular case c = its path is a straight line 
perpendicular to the force. The problem is an illustration of the theory of ' gyrostatic 
systems,' to be referred to in Chapter vi. 

70. The formula (1) of Art. 67, as amended by the addition of the teim 
A log z, may readily be generalized so as to apply to any case of irrotational 
motion in a region with circular boundaries, one of which encloses all the rest. 
In fact, for each internal boundary we have a series of the form 

Abg(,-o) + ^L + ( -^ + ..... (1) 

* Bayleigh, "On the Irregular Flight of a Tennis Ball," Mess, of Math. vii. (1878) [Papers, 
i. 344]; Greenhill, Mess, of Math. ix. 113 (1880). 
t Greenhill, I.e. 



69-7o] Transformations 81 

where c, = a + ib say, refers to the centre, and the coefficients A, A ly A 2 , ... 
are in general complex quantities. The difficulty however of determining these 
coefficients so as to satisfy given boundary conditions is now so great as to 
render this method of very limited application. 

Indeed the determination of the irrotational motion of a liquid subject to 
given boundary conditions is a problem whose exact solution can be effected 
by direct processes in only a limited number of cases. When the boundaries 
consist of fixed straight walls, a method of transformation devised by Schwarz * 
and Christoffelf , to be explained in Art. 73, is available. Most of the problems 
however whose solution is known have been obtained by an inverse method, 
viz. we take some known form of or yfr and inquire what boundary conditions 
it can be made to satisfy. Some simple examples of this procedure have already 
been given in Arts. 63, 64. 

If we take a known problem of flow with given fixed boundaries, where 
w=f(z), say, and apply a conformal transformation z = x( z ')> the transformed 
boundaries in the plane of z will still be stream -lines, and in this way we 
derive the solution of a new problem. It is sometimes advantageous to effect 
the transformation in two or more successive steps. 

A problem which has led to important transformations in this way is that of the 
flow past a fixed circular cylinder. It is easily seen from Arts. 68, 69 that the general 
solution of this is 

w=£ ^ + ^_ iF (*-f) + |logi, (2) 

where — U, — V are the component velocities at infinity, and k is the circulation. The 
procedure followed is to write 

z=t + c, (3) 

where t is an intermediate complex variable and | c |<a, and finally 

b 2 
*' = ' + 7 (4) 

It is obvious that the infinitely distant regions of the planes z and z' will be identical, and 
the general direction of the stream, and the value of the circulation, therefore the same. 
The constants c and b are adjusted so that the points t=±b in the plane of t may corre- 
spond to two arbitrary points A, B in the plane of z. 

For instance, let AB be a chord of the circle r = a, parallel to Ox and subtending an 
angle 2/3 at the centre 0. Referring to the figure on the next page we find 

c=-mcos/3, 6 = asin/3 (5) 

* "Ueber einige Abbildungsaufgaben," Crelle, lxx. [Gesammelte Abhandlungen, Berlin, 1890, 
ii. 65]. 

t "Sul problema delle temperature stazionarie e la rappresentazione di una data superficie," 
Ann. di. Mat. (2) i. 89. See also Kirchhoff, " Zur Theorie des Condensators," Berl. Monatsber. 
1877 [Ges. Abh. 101]. Many of the solutions which can thus be obtained have interesting applica- 
tions in Electrostatics, Heat-Conduction, &c. See, for example, J. J. Thomson, Recent 
Researches in Electricity and Magnetism, Oxford, 1893. 



82 Motion of a Liquid in Two Dimensions [chap, iv 

Then if P be any other point in the plane of z we have 

z = OP, t=CP (6) 

It follows from (4) that i^ = 0^|V (7) 

Writing for a moment 

t-b = r l e i0 ^ 1 t + b=r 2 e i \ z'-2b=r 1 'e ie i i z'+2b = r 2 'e i0 *\ (8) 

we have 6{-6{=Z (0, - 2 ) (9) 




Z' 




Now let P describe the circle in the plane of 0, in the positive direction, starting from A. 
The corresponding point P' in the plane of z' will, by (9), move so that the angle A'P'B' is 
constant and equal to 2/3, the path therefore being an arc of a circle. As P passes B, B 2 
increases by ir ; hence in order that the equation (9) may subsist, 6 2 must increase by 2tt. 
Hence as P completes its circle, P' moves back again along the arc B'A'. We thus obtain 
the case of a stream flowing in an arbitrary direction and with arbitrary circulation past a 
cylindrical lamina whose section is an arc of a circle* 

S-S/O-?). < 10 > 

the velocity at the edges A', B' will be infinite. It can be made finite, however, at one 
edge, say B', by a suitable determination of the circulation, viz. 

K =47ra(tf r cos/3- F sin/3) (11) 

The flow at B' is then given by 

u-iv=(Usinp+ V cos 0) sin pe 2ifi , (12) 

and is of course tangential to the arc. If the general velocity of the stream is W, at an 
inclination a to B'A', we have 

C7=-Fcosa, F=-Tfsina (13) 

Also, if R is the radius of the arc, 

asin/3 = i2sin2/3 (14) 

The ' lift,' therefore, at right angles to the stream, as given by Art. 72 b, is 

4 ^ 2 ^Sf cos(a+£) (15) 

If instead of the circle r = a in the figure we take as the circle to be transformed a circle 
touching it at A, and just including B, we get the profile of a Joukowsky aerofoil, of 

* Kutta, I.e. ante p. 79. Some related problems are discussed by Blasius, Zeitschr. f. Math. u. 
Phys. lix. 225 (1911). 



7o-7 i] General Problem of Translation 83 

which the circular arc is, as it were, the skeleton* This has a cusp at the point corre- 
sponding to A, and so involves an infinite velocity at this point (only). This singularity 
may be avoided by giving a suitable value to k. 

A simple method of obtaining solutions in two important cases of two- 
dimensioned motion is explained in the following Arts. 

71. Case I. The boundary of the fluid consists of a rigid cylindrical 
surface which is in motion with velocity U in a direction perpendicular to the 
length. 

Let us take as axis of x the direction of this velocity U, and let 8s be an 
element of the section of the surface by the plane xy. 

Then at all points of this section the velocity of the fluid in the direction 
of the normal, which is denoted by dyjr/ds, must be equal to the velocity of 
the boundary normal to itself, or — Udy/ds. Integrating along the section, 
we have 

ijr = — Uy 4- const (1) 

If we take any admissible form of yjr, this equation defines a system of curves 
each of which would by its motion parallel to x give rise to the stream-lines 
yfr = const, f. We give a few examples. 

1°. If we choose for yjr the form — Uy, (1) is satisfied identically for all 
forms of the boundary. Hence the fluid contained within a cylinder of any 
shape which has a motion of translation only may move as a solid body. 
If, further, the cylindrical space occupied by the fluid be simply-connected, 
this is the only kind of irrotational motion possible. This is otherwise evident 
from Art. 40; for the motion of the fluid and the solid as one mass evidently 
satisfies all the conditions, and is therefore the only solution which the problem 
admits of. 

2°. Let y\r = Ajr . sin 6; then (1) becomes 

A 



r 



sin# = — Ur sin 6 = const (2) 



In this system of curves is included a circle of radius a, provided Aja = — Ua. 
Hence the motion produced in an infinite mass of liquid by a circular cylinder 
moving through it with velocity U perpendicular to its length, is given by 



Ua 2 . A 
^ = -— sintf, (3) 



which agrees with Art. 68. 



* For further developments, and modifications of the method, reference may be made to 
Glauert, Aerofoil and Airscrew Theory, Cambridge, 1926. 

t Cf. Eankine, I.e. ante p. 63, where the method is applied to obtain curves resembling the 
lines of ships. 



84 Motion of a Liquid in Two Dimensions [chap, iv 

3°. Let us introduce the elliptic co-ordinates f , 77, connected with x, y by 

the relation 

x-\-iy — o cosh (f + ir)), (4) 

or x = c cosh f cos 77,) 

y = c sinh f sin t?*] ' ^ ' 

(cf. Art. 66), where f may be supposed to range from to 00 , and 77 from to 

27r. If we now put 

+ t> = Ce-G+*fl, (6) 

where is some real constant, we have 

yfr = — Ce~Z sin 77, (7) 

so that (1) becomes Ce - ^ sin rj=Uc sinh f sin 7; + const. 




In this system of curves is included the ellipse whose parameter £ is 

determined by 

Ce~Zo = £/csinh f . 

If a, b be the semi-axes of the ellipse we have 

a = c cosh f , 6 = c sinh £ , 

so that = r = lib r . 

a — 6 \a — b/ 

Hence the formula yjr=z-JJb( j) e~^sinr) (8) 



71] Translation of an Elliptic Cylinder 85 

gives the motion produced in an infinite mass of liquid by an elliptic cylinder 
of semi-axes a, 6, moving parallel to the greater axis with velocity U. 

That the above formulae make the velocity zero at infinity appears from 
the consideration that, when f is large, Bx and By are of the same order as 
e^SI; and e^Brj, so that dyfr/dx, dyfr/dy are of the order e~ 2 % or 1/r 2 , ultimately, 
where r denotes the distance of any point from the axis of the cylinder. At 
infinity yfr tends to the form A sin 6/r as in the case of a double source. 

If the motion of the cylinder were parallel to the minor axis, the formula 
would be 

ir^Va^^fe-ioosv (9) 

The stream-lines are in each case the same for all confocal elliptic forms 
of the cylinder, so that the formulae hold even when the section reduces to 
the straight line joining the foci. In this case (9) becomes 

yjr= VceScosr), (10) 

which would give the motion produced by an infinitely long lamina of breadth 
2c moving 'broadside on' in an infinite mass of liquid. Since however this 
solution makes the velocity infinite at the edges, it is subject to the practical 
limitation already indicated in several instances*. 

The kinetic energy of the fluid is given by 

2T = p Udylr = pC 2 e- 2 £ol 2n cos 2 7) drj 

= Trpb 2 U\ (11) 

where b is the half-breadth of the cylinder perpendicular to the direction of 
motion. 

Where there is circulation k round the cylinder we have merely to add to 
the above values of ojr a term /cf/27r. In the case of the lamina the value of k 
may be adjusted so as to make the velocity finite at one edge, but not at both. 
If the units of length and time be properly chosen we may write for (4) and (6) 
x+ iy = cosh (£ + irj), <t> + fy = e~ { ^ +il,) , 

whence x =* ( l+ ^hp) ' *-+ i 1 -0^p) • 

These formulae are convenient for tracing the curves cf> — const., ^ = const., which are 
figured on the preceding page. 

By superposition of the results (8) and (9) we obtain, for the case of an elliptic cylinder 
having a velocity of translation whose components are U, V, 

^ = ~(^r|) e~^ (Ub sin rj-Va cost}) (12) 

To find the motion relative to the cylinder we must add to this the expression 

Uy— Vx=c(U sinh £sin?7— Fcosh £cos?7) (13) 

* This investigation was given in the Quart. Journ. of Math. xiv. (1875). Eesults equivalent 
to (8), (9) had however been obtained, in a different manner, by Beltrami, "Sui principii fonda- 
mentali dell' idrodinamica razionale," Mem. deW Accad. delle Scienze di Bologna, 1873, p. 394. 
[Opere matematiche, Milano, 1904, ii. 202.] 



86 Motion of a Liquid in Two Dimensions [chap, iv 

For example, the stream-function for a current impinging at an angle of 45° on a plane 
lamina whose edges are at x= ±c is 

^=- -^^ocsinh^cosiy-sin^), (14) 

where q is the velocity at infinity. This immediately verifies, for it makes ^ = for £ = 0, 
and gives 

for £=oo . The stream-lines for this case (turned through 45° for convenience) are shewn 
below. They will serve to illustrate some results to be obtained later in Chapter vi. 




If we trace the course of the stream-line \fs=0 from <f>= +oo to 0= — oo , we find that 
it consists in the first place of the hyperbolic arc »/ = J 7r, meeting the lamina at right angles; 
it then divides into two portions, following the faces of the lamina, which finally re-unite 
and are continued as the hyperbolic arc ?7 = f 7r. The points where the hyperbolic arcs 
abut on the lamina are points of zero velocity, and therefore of maximum pressure*. It is 
plain that the fluid pressures on the lamina are equivalent to a couple tending to set 
it broadside on to the stream ; and it is easily found that the moment of this couple, per 
unit length, is ^7rpqd 2 c 2 f. Compare Art. 124. 

72. Case II. The boundary of the fluid consists of a rigid cylindrical 
surface rotating with angular velocity co about an axis parallel to its length. 

Taking the origin in the axis of rotation, and the axes of x, y in a perpen- 
dicular plane, then, with the same notation as before, d\fr/ds will be equal to 
the normal component of the velocity of the boundary, or 

dyfr _ dr 
ds ds' 

* Prof. Hele Shaw has made a number of beautiful experimental verifications of the forms of 
the stream-lines in cases of steady irrotational motion in two dimensions, including those figured 
on p. 78 and on this page; see Trans. Inst. Nav. Arch. xl. (1898). The theory of his method will 
find a place in Chapter xi. 

t When the general direction of the stream makes an angle a with the lamina the couple is 
%irpq 2 c 2 sin 2a. Cisotti, Ann. di. mat. (3), xix. 83 (1912). 



71-72] Rotating Boundary 87 

if r denote the radius vector from the origin. Integrating we have, at all points 

of the boundary, 

yfr = £&>r 2 + const (1) 

If we assume any possible form of yjr, this will give us the equation of a series 

of curves, each of which would, by rotating round the origin, produce the 

system of stream-lines determined by \fr. 

As examples we may take the following: 

1°. If we assume yfr = Ar 2 cos 20 = A {x 2 - y 2 ), (2) 

the equation (1) becomes 

(iG>-A)x 2 + (ia> + A)y 2 = C, 
which, for any given value of A, represents a system of similar conies. That 
this system may include the ellipse 

a 2 *b 2 ~ ' 
we must have (\a> — A) a 2 = (\w + A)b 2 , 

or A = |q> . = . 

* or + 6 2 

a 2 - b 2 
Hence the formula \fr = \ w . 2 2 (^g 2 — y 2 ) (3) 

gives the motion of a liquid contained within a hollow cylinder whose section 
is an ellipse with semi-axes a, 6, produced by the rotation of the cylinder 
about its longitudinal axis with angular velocity &>. The arrangement of the 
stream-lines yfr = const, is shewn on the next page. 
The corresponding formula for <j> is 

a 2 — b 2 

+—»-#+v-*y (*) 

The kinetic energy of the fluid, per unit length of the cylinder, is given by 

This is less than if the fluid were to rotate with the boundary, as one rigid 

mass, in the ratio of 

f a 2 - b 2 \ 2 

U 2 + 6V 

to unity. We have here an illustration of Lord Kelvin's minimum theorem, 

proved in Art. 45. 

2°. With the same notation of elliptic co-ordinates as in Art. 71, 3°, let 

us assume 

<j> + i,lr = Cie- 2 ti +il » (6) 

Since oc 2 -\-y 2 = \c 2 (cosh 2f + cos 2rj), 

the equation (1) becomes 

Ce~^ cos 2rj — \wc 2 (cosh 2f + cos 2r)) = const. 



88 Motion of a Liquid in Two Dimensions [chap, iv 

This system of curves includes the ellipse whose parameter is f , provided 

or, using the values of a, b already given, 

a=Jo)(a + 6) 2 , 
so that yfr = \(o (a + b) 2 e~ 2 ^ cos 2rj, 



cf> = lo»(a + b) 2 e- 2 tsm2 v .\ ^ 

At a great distance from the origin the velocity is of the order 1/r 3 . 

The above formulae therefore give the motion of an infinite mass of liquid, 
otherwise at rest, produced by the rotation of an elliptic cylinder about its 
axis with angular velocity co* The diagram shews the stream-lines both 
inside and outside a rigid elliptical cylindrical case rotating about its axis. 




The kinetic energy of the external fluid is given by 

2T=l7rpc 4 .a> 2 (8) 

It is remarkable that this is the same for all confocal elliptic forms of the 
section of the cylinder. 

Combining these results with those of Arts. 66, 71 we find that if an 
elliptic cylinder be moving with velocities U, V parallel to the principal axes 
of its cross-section, and rotating with angular velocity <w, and if (further) the 

* Quart. Journ. Math. xiv. (1875) ; see also Beltrami, I.e. ante p. 85. 



72] Rotating Cylinder 89 

fluid be circulating irrotationally round it, the cyclic constant being te, then 
the stream-function relative to the aforesaid axes is 

yfr = - /(5Lt|W* ( Ub sin v - Va cos V ) + J« (a + 6) 2 <r 2 * cos 2t; + ^- f . 

(9) 

The ^>a^5 followed by the particles of fluid in several of the preceding 
cases, as distinguished from the stream-lines, have been studied by Prof. 
W. B. Morton*; they are very remarkable. The particular case of the circular 
cylinder (Art. 68) was examined by Maxwell")*. 

3°. Let us assume \js = Ar 3 cos 30 = A(x 3 - Zxy 2 ). 

The equation (1) of the boundary then becomes 

A (x 3 -3xy 2 )-%a>(x 2 +f) = C. (10) 

We may choose the constants so that the straight line x=a shall form part of the boundary. 
The conditions for this are 

^a 3 -£o>a 2 = <7, 34a+£a> = 0. 

Substituting in (10) the values of A, C hence derived, we have 
x 3 - a 3 - Zxy 2 + 3a (x 2 - a 2 +y 2 ) = 0. 
Dividing out by x - a, we get x 2 + 4ax + 4a 2 - 3y 2 , 

or x + 2a=± f J3.y. 

The rest of the boundary consists therefore of two straight lines passing through the point 
( - 2a, 0), and inclined at angles of 30° to the axis of x. 

We have thus obtained the formulae for the motion of the fluid contained within a 
vessel in the form of an equilateral prism, when the latter is rotating with angular velocity 
to about an axis parallel to its length and passing through the centre of its section; viz. 
we have 

^-^-r 3 cos 30, (^=^-^8^30, (11) 

where 2 v/3a is the length of a side of the prism J. 

4°. In the case of a liquid contained in a rotating cylinder whose section is a circular 
sector of radius a and angle 2a, the axis of rotation passing through the centre, we may 
assume 

COS 20 /r\ (2n+l)7r/2a -r/J 

^=i^ 2 ^ + ^ 2n + 1 (M cos(2* + l)£, (12) 



cos 2a ' n + 1 \aJ v T ; 2a 

the middle radius being taken as initial line. For this makes \js — \ ar 2 for 6 — ± a, and the 
constants A 2n+1 can be determined by Fourier's method so as to make yjr—^coa 2 for r=a. 
We find 

^n + i-(-)- + 1 a>a 2 { (2 ^^_ 4a - ( ^^ + ( ^ + l ^ + 4a } (13) 

The conjugate expression for <$> is 

, , 2 sin20 . / r \ (2/i+D w /2a . n Q 
♦--^SHT^-^-W ^(2^+1)- (14) 

* Proc. Boy. Soc. A, lxxxix. 106 (1913). 

f Proc. Lond. Math. Soc. iii. 82 (1870) [Papers, ii. 208]. 

X The problem of fluid motion in a rotating cylindrical case is to a certain extent mathe- 
matically identical with that of the torsion of a uniform rod or bar. The examples numbered ' 1° ' 
and '3°' are mere adaptations of two of de Saint- Venant's solutions of the latter problem. See 
Thomson and Tait, Art. 704 et seq. 



90 Motion of a Liquid in Two Dimensions [chap, iv 

The kinetic energy is given by 

2T= - p U^ds= -2p<o ("fardr, (15) 

where cp a denotes the value of cp for B — a, the value of dcp/dn being zero over the circular 
part of ihe boundary*. 

The case of the semicircle <x = \tt will be of use to us later. We then have 

^■W M 7fe-W + Ri' (16) 

and therefore 

f a 6 rdr=°^2 -i-|-i 2 , l } = «* U «*\ 

J Va rr 2n + 3\2n-l 2n + l^ 2n + ZJ tt \ 8/' 

Hencet 2T=$irp<o 2 a i C^-^ = '3106a 2 x|7rpa> 2 a 2 (17) 

This is less than if the fluid were solidified, in the ratio of 6212 to 1. Cf. Art. 45. 

72 a. We have seen in several instances that when a cylinder has a motion 
of translation though an infinite fluid the effect at a great distance is that of 
a double source. A general formula for this can be given in terms of certain 
constants which occur in the expression for the kinetic energy of the fluid j. 

If we write <p=Ufa + Vfa, (1) 

where ( U, V) is the velocity of the cylinder, the functions fa , fa are determined by the 
conditions that V 2 fa = 0, v 2 2 = O throughout the external space, that their derivatives 
vanish at infinity, and that at the contour of the cylinder 

_dp -*tt-n, (2) 

on on 

where (I, m) is the direction of the outward normal. Hence the energy of the fluid is given 

by 

— = - L^ ds=AU 2 + 2HUV+BV 2 , (3) 

p J on 

where A = - I fa -^ ds= llfads, \ 

B= — I fa ~-ds= jmfads, V 

H= — J fa -+^ds= - I fa ~ds= lmfads= Jlfads. 

The two forms of E are equal by the two-dimensional form of Green's Theorem. Cf. Art. 
121, where the general three-dimensional case is discussed. 

Referring to Art. 60 (7), suppose that a cylinder of any form of section is moving with 
unit velocity parallel to the axis of x. Taking an origin within the contour, and writing 

r 2 = {xQ~xf + {y -yf 

= r 2 - 2(^o+3/3/o) +•••> ( 5 ) 

* This problem was first solved by Stokes, "On the Critical Values of the Sums of Periodic 
Series," Camh. Trans, viii. (1847) [Papers, i. 305]. See also Hicks, Mess, of Math. viii. 42 (1878) ; 
Greenhill, ibid. viii. 89, and x. 83. 

t Greenhill, I.e. J Cf. Proc. Boy. Soc. A, cxi. 14 (1926) and Art. 300 infra. 



•(4) 



2-72 b] Source due to a Moving Cylinder 91 



where (x 0i y ) is a distant point at which the value of cj> is required, and (#, y) a point of 
the contour, we have 

log,=logr -^ + »+... > (6) 

and _ (log ,. )= _^_ r A0 ) 

approximately. Writing 

<£ = $!, <p'=-x (7) 

in the formula referred to, we find 

(A + Q)x Q +Hyp /fi v 

^ p== V ' ^ 

where 4 and 5" are defined by (4), and 

Q=jlxds, (9) 

i.e. Q denotes the sectional area of the cylinder. 

The flow at a great distance is accordingly that due to a double source, but the axis of 
the source does not in general coincide with the direction of motion of the cylinder. 

The generalization of (8) is obvious. When the cylinder has a velocity ( £7, V) we have 

2*r *<t> P ={(A + Q) U+ffV}x + {HU+(B+Q)V}y () (10) 

In terms of the complex variables w, z, this may be written 

W = (a + i(3)/z , (11) 

with 2rra = (A + Q)U+HV, 2irP=HU+(B+Q)V. (12) 

For an elliptic cross-section we have, by comparison of Art. 71 (11) with (3) above, 
A = 7rb 2 , B = 7ra 2 , whilst Q = nab. Hence 

<!> P =(a + b)(bUxo + aVy Q )l2r * (13) 

72 b. The hydrodynamic forces on a fixed cylinder due to the steady 
irrotational motion of a surrounding fluid have already been calculated in 
one or two cases. A general method, available whenever the form of w, 
— (f> + iyfr, for the fluid motion is known, has been given by Blasius *. 

The pressures on the contour may be reduced to a force (X, Y) at the 
origin, and a couple N. If 6 be the angle which the velocity q makes with 
the axis of x, we have 

Y + iX = - ipjq 2 (cos - ism6)ds, (1) 

where the integral is taken round the contour of the cylinder. 
This may be written 

Y+iX^-yfiqe-vf^ds^-ipf^jdz, (2) 

This gives X and Y. 

Again, if S- be the angle which an element Bs of the contour makes with 
the radius vector (produced), 

N = fpr cos ^rds=jprdr = - \pj{u 2 + v 2 )(xdx + ydy), (3) 

* " Funktiontheoretische Methoden in der Hydrodynamik," Zeitschr. f. Math. u. Phys. Iviii. 
(1910). 



92 Motion of a Liquid in Two Dimensions [chap, iv 

Now along a stream-line we ha,ve.vdx = udy, whence 

(u - ivf (dx + idy) = (u 2 + v 2 ) (dx - idy) 
and 

(u — ivf (x + iy) (dx + idy) = (u 2 + v 2 ) [xdx + ydy + i (y dx - xdy)\, 

Hence N is given by the real part of the integral 



■/( 



£)'•«•- <« 



In the case of a cylinder immersed in a uniform stream, with circulation, 
the value of w at a great distance tends to the form 

w = A + Bz + Clog z (5) 

Since in (4) there are no singularities of the integrand in the space occupied 
by the fluid, the integral may be replaced by that round an infinite enclosing 
contour. On this understanding 

If the stream at infinity is ( U, V), and if k denote the circulation, we have 

B = -(U-iV), G=-%k\1tt (7) 

Hence X = icpV, Y=-k P U (8) 

which is the generalization of the result obtained in Art. 69 for the particular 
case of a circular section. 

For the calculation of the moment N the expression in (5) must be carried 
a stage further. Writing 



w- A + Bz+C\ogz + — , (9) 

we have 

'g) ! =* + ^ + ^5 + (10) 

Omitting all the terms which disappear in the case of an infinite contour we 
have 



c, 



/(^) 2 zdz = 2>rri(C 2 -2BD) (11) 



Substituting the values of B and G from (7), writing D = a + iff, and taking 
the real part, we find 

N=2irp{ffU-aV) (12) 

If by the superposition of a general velocity (— U, — V) the fluid were 
reduced to rest at infinity, the term D/z in (9) would be due to a translation 
of the cylinder with this velocity. Hence the values of a, {3 are as given in 
Art. 72, except that the signs are reversed. Hence (12) gives 

N= P {(A-B)UV-H(U 2 -V 2 )} (13) 

Thus for an elliptic section referred to its principal axes 

N = -irp{a 2 ~b 2 ) UV. (14) 



72 b] Blasius' Theorems 93 

As a further application of Blasius' formula we may calculate the force on a fixed 
cylinder due to an external source. 

We write w= — plog(z — c) +/(«), (15) 

where the first term represents the source at z = c, say, and f(z) its image in the cylinder, 
i.e. f{z) is the addition necessary to annul the normal velocity at the contour, due to the 
source. Hence 

S--A*™ ™ 

The contour integral in (2) is now equal to the integral round an infinite contour minus the 
integral (in the positive sense) round the singularity at z = c. The infinite contour gives a 
zero result. In the neighbourhood of the singularity the only part of (dwjdz) 2 which need 
be taken into account is that containing the first power of z — c in the denominator, viz. 

Ml ^ 

z-c ' 

ultimately. Hence Y+iX= - Mppf (c) (17) 

The form of / (z) for the case of a circular cylinder is already known from Art. 64, 3°. 
The source being supposed on the axis of x, so that c is real, we have 

f(z)=-p\og(z-a 2 /c) + p\ogz, (18) 

f'^-c^y • < 19 > 



*-^5F^ r =° ( 2 °)* 



27r/z 2 a 2 
c{c 2 -a 2 ) 

In the general case, an approximation to the asymptotic form which f(z) assumes, when 
the distance of the source is great compared with the dimensions of the cross-section, is 
obtained if we suppose it to represent the effect of a translation of the cylinder with a 
velocity equal and opposite to that which the source would produce in the neighbourhood, 
if the cylinder were absent. Thus, the source being still assumed to be on the axis of x, 
we have from Art. 72 a 

m jA±3^m, (21) 

where tf=/»/& Hence f( c ) = JA±^±^lt > (22 ) 

and therefore x JA + Q)y?P ^ Y—?& (23) 

Iff(=p 2 /(?) is the acceleration at the position of the origin, in the undisturbed stream, 
these results may be written 

X=p(A + Q)f, Y=- P Hf. (24) 

For a circular section A = 7ra 2 , H—Q, Q=7ra 2 , and the formula (20) is verified, if we 
neglect terms of the order a 2 \c 2 . 

A number of elegant applications of Blasius' method, relating to the mutual action of 
circular cylinders, with circulation, have been made by Cisottit. One of his results may 
be quoted. A cylinder of radius b is fixed excentrically within a cylindrical tunnel of 
radius a, and the intervening space is occupied by fluid having a circulation k. The 
resultant force on the cylinder is towards the nearest part of the tunnel wall, and has the 
value 

K 2 d 2 +27ry/{(a + b + d)(a + b-d)(a-b + d)(a-b-d)}, 

where d is the distance between the axes. 

* The result is due to Prof. G. I. Taylor. f Rend. d. r. Accad. d. Lincei (6) i. (1925-6). 



94 Motion of <% Liquid in Two Dimensions [chap, iv 

Free Stream- Lines. 

73. The first solution of a problem of two-dimensional motion in which 
the fluid is bounded partly by fixed plane walls and partly by surfaces of 
constant pressure, was given by Helmholtz*. Kirchhofff and others have 
since elaborated a general method of dealing with such questions- If the 
surfaces of constant pressure be regarded as free, we have a theory of jets, 
which furnishes some interesting results in illustration of Art. 24. Again 
since the space beyond these surfaces may be filled with liquid at rest, with- 
out altering the conditions of the problem, we obtain also a number of cases 
of 'discontinuous motion,' which are mathematically possible with perfect 
fluids, but whose practical significance is more open to question. We shall 
return to this point at a later stage (Chap. XI.); in the meantime we shall 
speak of the surfaces of constant pressure as 'free.' Extraneous forces, such 
as gravity, being neglected, the velocity must be constant along any such 
surface, by Art. 21 (2). 

The method in question is based on the properties of the function ? 
introduced in Art. 65. The moving fluid is supposed bounded by stream- 
lines yjr = const., which consist partly of straight walls, and partly of lines 
along which the resultant velocity (q) is constant. For convenience, we may 
in the first instance suppose the units of length and time to be so adjusted 
that this constant velocity is equal to unity. Then in the plane of the 
function f the lines for which q = 1 are represented by arcs of a circle of unit 
radius, having the origin as centre, and the straight walls (since the direction 
of the flow along each is constant) by radial lines drawn outwards from the 
circumference. The points where these lines meet the circle correspond to 
the points where the bounding stream-lines change their character. 

Consider, next, the function log f. In the plane of this function the 
circular arcs for which q = 1 become transformed into portions of the 
imaginary axis, and the radial lines into lines parallel to the real axis, since 
if ^—q~ x e iB we have 

log£=logi + t0 (1) 

It remains, then, to determine a relation of the formj 

log?=/(w), (2) 

where tv = (f> + iyfr, as usual, such that the rectilinear boundaries in the plane 

of log f shall correspond to straight lines -v/r = const, in the plane of w. 

There are further conditions of correspondence between special points, one 

on the boundary, and one in the interior, of each region, which render the 

problem determinate. 

* Loc. cit. ante p. 75. 

t "Zur Theorie freier Fliissigkeitsstrahlen," Crelle, lxx. (1869) [Ges. Abh. p. 416]. See also 
his Mechanik, cc. xxi., xxii. 

% The use of log £, in place of f, is due to Planck, Wied, Ann. xxi. (1884). 



73] Free Stream-Lines 95 

When the correspondence between the planes of f and w has been 
established, the connection between z and w is to be found, by integration, 
from the relation 

£— {■ (3) 

aw 

The arbitrary constant which appears in the result is due to the arbitrary 
position of the origin in the plane of z. 

The problem is thus reduced to one of conformal representation between 
two areas bounded by straight lines*. This is resolved by the method of 
Schvvarz and Christoffel, already referred tof, in which each area is repre- 
sented in turn on a half-plane. Let Z (= X + iY) and t be two complex 
variables connected by the relation 

^ = A(a-t)-^'(b-t)-^'(c-t)-yl ir ..., (4) 

where a, b, c, . . . are real quantities in ascending order of magnitude, whilst 
a, /3, 7, ... are angles (not necessarily all positive) such that 

a + /3 + 7 + ... = 2tt; (5) 

and consider the line made up of portions of the real axis of t with small 
semi-circular indentations (on the upper side) about the points a, b, c, ... . 
If a point describe this line from t = — oo to t = -\- oo , the modulus only of the 
expression in (4) will vary so long as a straight portion is being described, 
whilst the effect of the clockwise description of the semi-circular portions is 
to introduce factors e ia , e ifi , e iy , ... in succession. Hence, regarding dZ/dt as an 
operator which converts 8t into BZ, we see that the upper half of the plane of 
t is conformably represented on the area of a closed polygon whose exterior 
angles are a, /3, 7, ...,by the formula 

Z = A${a-t)-*l" (b -t)-V" (c-t)-*l" ...dt + B, (6) 

provided the path of integration in the £-plane lies wholly within the region 
above delimited. When a, b, c, ...,«, (3> 7, ... are given, the polygon is com- 
pletely determinate as to shape; the complex constants A, B only affect its 
scale and orientation, and its position, respectively. 

As already indicated, we are specially concerned with the conformal 
representation of rectangular areas. If a = @ = y = 8 = %ir, the formula (6) 
becomes 

Z = A ) s/{(a-t){b-t){c-t){d-t)\+ B (7) 

It is easily seen that the rectangle is finite in all its dimensions unless two at 
least of the points a, 6, c, d are at infinity. The excepted case is the one 

* See Forsyth, Theory of Functions, c. xx. 
t See the footnotes on p. 81 ante. 



96 Motion of a Liquid in Two Dimensions [chap, iv 



Z-A\-,? : +U 



specially important to us; the two finite points may then conveniently be taken 
to be t = + 1, so that 

dt_ 

= A cosh-^ + 5 (8) 

In particular, the assumption 

t — cosh -j-, (9) 

where k is real, transforms the space bounded by the positive halves of the 
lines F=0, Y=7rk, and the intervening portion of the axis of F, into the 
upper half of the plane t Cf. Art. 66, 1°. 

Again, if the two finite points coincide, say at the origin of t, we have 



Z=AJj + B = A \ogt + B (10) 



This transforms the upper half of the £-plane into a strip bounded by two 
parallel straight lines. For example, if 

t = e z l\ (11) 

where k is real, these may be the lines F= 0, Y—irk. 

74. As a first application of the method in question, we may take the 
case of a fluid escaping from a large vessel by a straight canal projecting 
inwards*. This is the two-dimensional form of Borda's mouthpiece, referred 
to in Art. 24. 

The boundaries of corresponding areas in the planes of ?, log f, and w, 
respectively, are easily traced, and are shewn in the figures f. It remains to 
connect the areas in the planes of log f and w each with the upper half-plane 
of an intermediate variable t. It appears from equations (8) and (10) of the 
preceding Art. that this is accomplished by the substitutions 

log? =4 cosh- 1 * + 5, w=C\ogt + D (1) 

We have here made the corners A, A' in the plane of log f correspond to 
t=± 1, and we have also assumed that £ = corresponds to w = — oo , as is 
evident on inspection of the figures. To specify more precisely the values of 
the cyclic functions cosh -1 1 and log t we will assume that they both vanish 
at t = 1, aud that their values at other points in the positive half-plane are 
determined by considerations of continuity. It follows that when t = — 1 the 
value of each function will be iir. At the points A' , A in the plane of log f, 

* This problem was first solved by Helmholtz, I.e. ante p. 75. 

t The heavy lines correspond to rigid boundaries, and the fine continuous lines to free surfaces. 
Corresponding points in the various figures are indicated by the same letters. 



73-74] 



B or da's Mouthpiece 



97 



we have, on the simplest convention, log f =0 and liir, respectively; whence, 
towards determining the constants in (1), we have 

= B, 2iir = iwA + B, 

so that log£=2cosh- 1 £ (2) 

Again, in the plane of w we take the line //' as the line -\Jr = 0; and if the 
final breadth of the issuing jet be 26, the bounding stream-lines will be 
ylr— ±b. We may further suppose that <£ = is the equipotential curve passing 
through A and A'. Hence, from (1), 

ib = i7rC + D, -ib = D, 



26 1 , ■* 
w — — log t — ib. 

IT 



so that w — — log t — ib (3) 

It is easy to eliminate t between (2) and (3), and thence to find the relation 

? 



1 

r 
i 


3 


A 






.) 










K 








^ 




r 

logt 
i 


A' 


B 




_ ji 






J 


i' 




B' 




w 

A 



-I' 



between z and w by integration, but the formulae are perhaps more convenient 
in their present shape. 

The course of either free stream-line, say A' I, from its origin at A', is now 
easily traced. For points of this line t is real and ranges from 1 to 0; we 
have, moreover, from (2), id = 2 cosh -1 1, or t = cos \Q. Hence, also, from (3), 



(f> = — log cos \6. 



(4) 



Since, along this line, we have d<j>/ds = — q = — 1, we may put </> = — s, where 
the arc s is measured from A'. The intrinsic equation of the curve is 
therefore 



26 1 in 

s = — log sec \v. 

7T 



(5) 



98 



Motion of a Liquid in Two Dimensions [chap, iv 



From this we deduce in the ordinary way 

a? = — (sin 2 |0-logsec£0), v = -(0-sin0), (6) 

7T IT 

if the origin be at A'. By giving a series of values ranging from to it, the 
curve is easily plotted*. 




Line of Symmetry. 

Since the asymptotic value of y is b, it appears that the distance between 
the fixed walls is 46. The coefficient of contraction is therefore J, in accord- 
ance with Borda's theory. 

75. The solution for the case of fluid issuing from a large vessel by an 
aperture in a plane wall is analytically very similar. The chief difference is 
that the values of log f at the points A, A' in the figures must now be taken 

. ?■ _ 



Ir- 



J 
log}; 



A' 



A> B' 

to be and — iir, respectively, whence, to determine the constants A, B in. 
(1) we have Q^iwA+B, -iir = B, 

so that log f= cosh -1 t — iir (7) 

* To correspond exactly with p. 97 the figure should be turned through 180°. 



74- 



'6] Vena Contracta 99 



The relation between w and t is exactly as before, viz. 

w = — log t — ib, (8) 

7T 

where lb is the final breadth of the stream, between the free boundaries. 

For the stream-line AI, t is real, and ranges from — 1 to 0. Since, also, 
id = cosh -1 1 — iir we may put t = cos (0 + it), where 6 varies from to — -J7T. 
Hence, from (8), with </> = — s, we have, for the intrinsic equation of the stream- 
line, 



= f£lo g (_ S ec0) (9) 

7T 



From this we find 



x = - sin 2 ^, y=-{logtan(i7r + l<9)-sin<9}, (10) 

7T 7T 

if the point A in the plane of z be taken as origin*. The curve is shewn (in 
an altered position) below. 



Line of Symmetry. 

The asymptotic value of x, corresponding to 6 = — J7r, is 26/-7T, the half width 
of the aperture is therefore (it + 2)6/7r, and the coefficient of contraction is 

7r/(7r + 2) = '611. 

76. In the next example a stream of infinite breadth is supposed to impinge 
directly on a fixed plane lamina, and thence to divide into two portions bounded 
internally by free surfaces. 

The middle stream-line, after meeting the lamina at right angles, branches 
off into two parts, which follow the lamina to the edges, and thence form the 

* This example was given by Kirchhoff (I.e.), and discussed more fully by Rayleigh, "Notes 
on Hydrodynamics," Phil. Mag. Dec. 1876 [Papers, i. 297]. 



100 



Motion of a Liquid in Two Dimensions [chap, iv 



free boundaries. Let this be the line yfr = 0, and let us further suppose that 
at the point of divergence we have <£ = 0. The forms of the boundaries in the 
various planes are shewn in the figures. The region occupied by the moving 




logZ 



w 



-—i 



w 



>r~ 



fluid now corresponds to the whole of the plane w, which must however be 
regarded as bounded internally by the two sides of the line yfr — 0, <f> < 0. 

With the same conventions as in the beginning of Art. 75, we have 

log f =cosh -1 £ -iir, (1) 



or 



*=-cosh(iogo=-i(r-J). 



■(2) 



The correspondence between the planes of w and t is best established by 
considering first the boundary in the plane of w~\ The method of Schwarz 
and Christoffel is then at once applicable. Putting a = — 7r, /8 = y = . . . = 0, 
in Art. 73 (4), we have 

iv-^iAP + B (3) 



dt 



= AL 



At I we have t = 0, w~ x = 0, so that B = 0, or (say) 



w = — 



.(4) 



To connect C (which is easily seen to be real) with the breadth (I) of the 
lamina, we notice that along C A we have ? = <f~\ and therefore, from (2), 



*--*(£ + «). ? = -*-V(« 2 -l), 



.(5) 



76] Impact of a Stream on a Lamina 101 

the sign of the radical being determined so as to make q = for t = — oo . 
Also, dx/d(j> = - 1/q. Hence, integrating along G A in the first figure we have 

'= 2 /::s^=- 4 Cl=- 4 °/:: w-< ■■■«» 

whence G= — — r (7) 

Along the free boundary AI, we have log f= id, and therefore, from (2) and 

£ = -cos0, </>=-asec 2 (9 (8) 

The intrinsic equation of the curve is therefore 

s=-L-,se<*0, (9) 

7T + 4 

where 6 ranges from to — \tt. This leads to 

21 
tf= 7jr ^(sec0 + i7r), 

y = j {sec tan - log (£ir + £0)}, 

the origin being at the centre of the lamina. 



.(10) 



Line of Symmetry. 



The excess of pressure on the anterior face of the lamina is, by Art. 23 (7), 
equal to ^p(l — q 2 ). Hence the resultant force on the lamina is 

p/>-rtg*--^/:| i g-«)j--^/;>-i)$-^ft 

(11) 

It is evident from Art. 23 (7), and from the obvious geometrical similarity 
of the motion in all cases, that the resultant pressure (P , say) will vary as 



102 



Motion of a Liquid in Two Dimensions [chap, iv 



the square of the general velocity of the stream. We thus find, for an arbitrary 
velocity q *, 

Po=^^pqo 2 .l = -^Op q( ?.l (12) 

77. If the stream be oblique to the lamina, making an angle a, say, with 
its plane, the problem is modified in the manner shewn in the figures. 



// 



/ 



'/ 



< ? , 



I 




4 


i' 






C 




w 








A' 

















IV 



I A' 



The equations (1) and (2) of the preceding Art. still apply; but at the point I we now 
have £=e -*(*-«), and therefore £ = cosa. Hence, in place of (4)t, 

C 

(13) 



(if— cos a) 2 ' " 

At points on the front face of the lamina, we have, since ^ -1 = |^|, 
1 



= ±*+V(* 2 -l), q=±t- */(?-!), 



,(14) 



where the upper or the lower signs are to be taken according as t% 0, i.e. according as the 
point referred to lies to the left or right of C in the first figure. Hence 

dx ,ldcj) 2C 

^ =± ^ = (^o^±V(* 2 -l)} (15) 

Between A' and C, t varies from 1 to oo , whilst between A and C the range is from 
— oo to -1. If we put 

1 - COS O COS 0) 



t= 



COS O - COS 0) 



the corresponding ranges of o> will be from ir to a, and from a to 0, respectively ; and we 
find 

dt cos a — cos to . , , //j9 _. sin a sin o) 

sincoao), ±v(£ 1) = ' 



(t - cos a) 



sin* a 



cos a — cos co 



* Kirchhoff, I.e. ante p. 94; Kayleigh, "On the Kesistance of Fluids," Phil. Mag. Dec. 1876 
[Papers, i. 287]. 

f The solution up to this point was given by Kirchhoff (Crelle, I.e.) ; the subsequent discussion 
is taken, with merely analytical modifications, from the paper by Kayleigh. 



76-77] 



Pressure on a Lamina 



Hence 

and therefore 



dx 
da> 



2(7 



sin^o 



— (1 — cos a cos co + sin a sin a>) sin co, 



103 

.(16) 



x = . . {2 cos co + cos a sin 2 o + sin a sin co cos co + (h n — co) sin a}, (17) 

sin 4 a 

where the origin has been adjusted so that x shall have equal and opposite values when 
o> = and co = 7r, respectively; i.e. it has been taken at the centre of the lamina. Hence, in 
terms of C, the whole breadth is 

A. -4- tt sitl n 

(18) 



, _ 4 + 7r sin o ~ 



sin 4 a 

The distance, from the centre, of the point (<o=o) at which the stream divides is 

_2coso(l+sin 2 a) + (|7r-a)sina , »a» 

4 -I- it sin a 

To find the total pressure on the front face, we have 



\ ? {\-?)dx=±\ P [~ q yj t dt=±^c>](?-\) 

= — ?™ • snv 2 co da 

sin 3 a 



dt 



(t-cosa) 



.(20) 



Integrated between the limits it and 0, this gives 7rpC/sin 3 a. Hence, in terms of I, and of 
an arbitrary velocity q of the stream, we find 



Pn = 



7r sin a 



.pqf.l (21) 

4 + 7rsina r *° 

To find the centre of pressure, we take moments about the centre of the lamina. Thus 

hpl(I—q 2 )xdx= — r-^- . / #sin 2 coc£co 
2 W «m 3 a J n 

rrpC 



„ C cos a 

X ^ ; 

sm°o sin 4 a 



(22) 



on substituting the value of x from (17). The first factor represents the total pressure; 
the abscissa x of the centre of pressure is therefore given by the second, or, in terms of 
the breadth, 

_ „ cos a 7 /OQ x 

a? = f — : — .1 (23) 

4 4 + 7rsina 

In the following table, derived from Rayleigh's paper, the column I gives the excess of 
pressure on the anterior face, in terms of its value when a = 90° ; whilst columns II and III 
give respectively the distances of the centre of pressure, and of the point where the stream 
divides, from the centre of the lamina, expressed as fractions of the total breadth*. 



a 


I 


II 


III 


90° 


1-000 


•000 


•000 


70° 


•965 


•037 


•232 


50° 


•854 


•075 


•402 


30° 


•641 


•117 


•483 


20° 


•481 


•139 


•496 


10° 


•273 


•163 


•500 



* For a comparison with experimental results see Rayleigh, I.e. and Nature, xlv. (1891) 
[Papers, iii. 491]. 



104 Motion of a Liquid in Two Dimensions [chap, iv 

78. An interesting variation of the problem of Art. 76 has been discussed 

by Bobyleff *. A stream is supposed to impinge symmetrically on a bent lamina 

whose section consists of two equal straight lines forming an angle. 

If 2a be the angle, measured on the down-stream side, the boundaries in the plane of £ 
can be transformed, so as to have the same shape as in the Art. cited, by the assumption 

provided A and n be determined so as to make £' = 1 when ^=e~ l ^ 1t ~ a \ and (' = e~ llt when 
C=e" i( ^ +a) . This gives 

A=e~ i{hn - a) , n = 2a/7r. 
On the right-hand half of the lamina, t will be negative as before, and since 2"" 1= |t|, 

i = {-* + V(* 2 -l)}», q = {-t-^-\)Y (24) 

Hence 



/^^'"^/"j-^ 

/j**"^/! 1 "'"*" 1 ^?""^^''"^" 1 * 



dt 



*V(* 2 -i)' 

dt 



These can be reduced to known forms by the substitution 
where a ranges from to 1. We thus find 



7i\ - ~dt = -l-2?i — — - da>=-I-n-n 2 I — — da>, 
CJ -«,q dt J (l+(o) 2 Jo l+o) 

^ q -£dt = -l + 2n — — rs d<» = - 1 - w + w 2 / — — eta. 

CJ-ao^dt J o(l+») 2 Jo 1+0) 



(7 
We have here used the formulae 



.(25) 



J (l+w) J ol+o) 

jo(l+o)) 2 2 Jol+o) 



/ 



where 1 > h > 0. 

Since, along the stream-line, ds/d(f) = — l/q, we have from (25), if b denote the half- 
breadth of the lamina, 

6= 4 + LV4f£^4 (27) 

The definite integral which occurs in this expression can be calculated from the formula 

Irfi^-^pRD+i*^-**)-***-**). ( 28 > 

where ¥ (m), = d/dm . log n (m), is the function introduced and tabulated by Gauss t. 
The normal pressure on either half is, by the method of Art. 76, 

^jJ-'fLjW^^f^^^rfc j,. ip c.-^-....(29) 

2H J -ooXq *J dt 2 r j 1+w 2 r sin^TT r tt sin a v y 

* Journal of the Russian Physico- Chemical Society, xiii. (1881) [Wiedemann's Beiblatter, vi. 
163]. The problem appears, however, to have been previously discussed in a similar manner by 
M. K6thy, Klausenburger Berichte, 1879. It is generalized by Bryan and Jones, Proc. Roy. Soc. 
A, xci. 354 (1915). 

+ " Disquisitiones generales circa seriem infinitam... ," Werke, Gottingen, 1870... , iii. 161. 



78-79] 



Bobyleff's Problem 



105 



The resultant pressure in the direction of the stream is therefore 

4a 2 



P C. 



.(30) 



Hence, for any arbitrary velocity q of the stream, the resultant pressure is 

P=py.p q0 *b, 



.(31) 



where L stands for the numerical factor in (27). 

For = ^77, we have L = 2 + ^7r, leading to the same result as in Art. 76 (12). 

In the following table, taken (with a slight modification) from Bobyleff' s paper, the 
second column gives the ratio PjPo of the resultant pressure to that experienced by a 
plane strip of the same area. This ratio is a maximum when a = 100°, about, the lamina 
being then concave on the up-stream side. In the third column the ratio of P to the 
distance (2b sin a) between the edges of the lamina is compared with \pq^. For values of a 
nearly equal to 180°, this ratio tends to the value unity, as we should expect, since the fluid 
within the acute angle is then nearly at rest, and the pressure-excess therefore practically 
equal to \pq^- The last column gives the ratio of the resultant pressure to that experienced 
by a plane strip of breadth 26 sin a, as calculated from (12). 



a 


PlPo 


P/pq Q 2 b sin a 


PIP sin a 


10° 


•039 


•199 


•227 


20° 


•140 


•359 


•409 


30° 


•278 


•489 


•555 


40° 


•433 


•593 


•674 


45° 


•512 


•637 


•724 


50° 


•589 


•677 


•769 


60° 


•733 


•745 


•846 


70° 


•854 


•800 


•909 


80° 


•945 


•844 


•959 


90° 


1-000 


•879 


1-000 


100° 


1-016 


•907 


1-031 


110° 


•995 


•931 


1-059 


120° 


•935 


•950 


1-079 


130° 


•840 


•964 


1-096 


135° 


•780 


•970 


1-103 


140° 


•713 


•975 


1-109 


150° 


•559 


•984 


1-119 


160° 


•385 


•990 


1-126 


170° 


•197 


•996 


1-132 



Discontinuous Motions. 

79. It must suffice to have given a few of the more important examples of 
steady motion with a free surface, treated by what is perhaps the most system- 
atic method. Considerable additions to the subject have been made by Michell*, 
Lovef, and other writers J. It remains to say something of the physical 

* "On the Theory of Free Stream-lines," Phil. Trans. A, clxxxi. (1890). 

+ " On the Theory of Discontinuous Fluid Motions in Two Dimensions," Proc. Camb. Phil. 
Soc. vii. (1891). 

J For references see Love, Encycl. d. math. Wiss. iv. (3), 97.... A very complete account of 
the more important known solutions, with fresh additions and developments, is given by Greenhill, 



106 Motion of a Liquid in Two Dimensions [chap, iv 

considerations which led in the first instance to the investigation of such 
problems. 

We have, in the preceding pages, had several instances of the flow of a 
liquid round a sharp projecting edge, and it appeared in each case that the 
velocity there was infinite. This is indeed a necessary consequence of the 
assumed irrotational character of the motion, whether the fluid be incom- 
pressible or not, as may be seen by considering the configuration of the equi- 
potential surfaces (which meet the boundary at right angles) in the immediate 
neighbourhood. 

The occurrence of infinite values of the velocity maybe afforded by supposing 
the edge to be slightly rounded, but even then the velocity near the edge will 
much exceed that which obtains at a distance great in comparison with the 
radius of curvature. 




In order that the motion of a fluid may conform to such conditions, it is 
necessary that the pressure at a distance should greatly exceed that at the 
edge. This excess of pressure is demanded by the inertia of the fluid, which 
cannot be guided round a sharp curve, in opposition to centrifugal force, 
except by a distribution of pressure increasing with a very rapid gradient 
outwards. 

Report on the Theory of a Stream-line past a Plane Barrier, published by the Advisory Committee 
for Aeronautics, 1910. 

The extension to the case of curved rigid boundaries is discussed in a general manner in various 
papers by Levi-Civita and Cisotti. For these, reference may be made to the Rend. d. Circolo Mat. 
di Palermo, xxiii. xxv. xxvi. xxviii. and the Rend. d. r. Accad. d. Lincei, xx. xxi. ; the working 
out of particular cases naturally presents great difficulties. The matter was treated later by 
Leathern, Phil. Trans. A, ccxx. 439 (1915) and H. Levy, Proc. Roy. Soc. A, xcii. 107 (1915). The 
theory of mutually impinging jets is treated very fully by Cisotti, " Vene confluenti," Ann. di 
mat. (3) xxiii. 285 (1914). 



79] Discontinuous Motions 107 

Hence, unless the pressure at a distance be very great, the maintenance of 
the motion in question would require a negative pressure at the corner, such 
as fluids under ordinary conditions are unable to sustain. 

To put the matter in as definite a form as possible, let us imagine the 
following case. Let us suppose that a straight tube, whose length is large 
compared with the diameter, is fixed in the middle of a large closed vessel 
filled with frictionless liquid, and that this tube contains, at a distance from 
the ends, a sliding plug, or piston, P, which can be moved in any required 
manner by extraneous forces applied to it. The thickness of the walls of the 
tube is supposed to be small in comparison with the diameter; and the edges, 
at the two ends, to be rounded off, so that there are no sharp angles. Let us 
further suppose that at some point of the walls of the vessel there is a lateral 
tube, with a piston P, by means of which the pressure in the interior can be 
adjusted at will. 

Everything being at rest to begin with, let a slowly increasing velocity 
be communicated to the plug P, so that (for simplicity) the motion at any 
instant may be regarded as approximately steady. At first, provided a 
sufficient force be applied to Q, a continuous motion of the kind indicated in 
the diagram on p. 74 will be produced in the fluid, there being in fact only 
one type of motion consistent with the conditions of the question. As the 
acceleration of the piston P proceeds, the pressure on Q may become 
enormous, even with very moderate velocities of P, and if Q be allowed to 
yield, an annular cavity will be formed at each end of the tube. 

It is not easy to make out the further course of the motion in such a case 
from a theoretical standpoint, even in the case of a 'perfect' fluid. In actual 
liquids the problem is modified by viscosity, which prevents any slipping of 
the fluid immediately in contact with the tube, and must further exercise 
a considerable influence on such rapid different motions of the fluid as are 
here in question. 

As a matter of observation, the motions of fluids are often found to 
differ widely, under the circumstances supposed in each case, from the types 
represented on such diagrams as those of pp. 73, 74, 84, 86. In such a case 
as we have just described, the fluid issuing from the mouth of the tube does 
not immediately spread out in all directions, but forms, at all events for some 
distance, a more or less compact stream, bounded on all sides by fluid nearly 
at rest. A familiar instance is the smoke-laden stream of gas issuing from a 
chimney. In all such cases, however, the motion in the immediate neighbour- 
hood of the boundary of the stream is found to be wildly irregular*. 

It was the endeavour to construct types of steady motion of a frictionless 

* Certain experiments would indicate that jets may be formed before the ' limiting velocity ' of 
Helmholtz is reached, and that viscosity plays an essential part in the process. Smoluchowski, 
" Sur la formation des veines d'efnux dans les liquides," Bull, de V Acad, de Cracovie, 1904. 



108 Motion of a Liquid in Two Dimensions [chap, iv 

liquid, in two dimensions, which should resemble more closely what is 
observed in such cases as we have referred to, that led Helmholtz* and 
Kirchhoff* to investigate the theory of free stream-lines. It is obvious that 
we may imagine the space beyond a free boundary to be occupied, if we 
choose, by liquid of the same density at rest, since the condition of constant 
pressure along the stream-line is not thereby affected. In this way the 
problems of Arts. 76, 77, for example, give us a theory of the pressure 
exerted on a fixed lamina by a stream flowing past it, or (what comes to the 
same thing) the resistance experienced by a lamina when made to move with 
constant velocity through a liquid which would otherwise be at rest. 

The question as to the practical validity of this theory will be referred to 
later in connection with some related problems (Chapter XI.). 

Flow in a Curved Stratum. 

80. The theory developed in Arts. 59, 60 may be readily extended to 
the two-dimensional motion of a curved stratum of liquid, whose thickness is 
small compared with the radii of curvature. This question has been discussed 
from the point of view of electric conduction, by Boltzmannf, KirchhoffJ, 
Topler§, and others. 

As in Art. 59, we take a fixed point A, and a variable point P, on the surface defining 
the form of the stratum, and denote by \jr the flux across any curve AP drawn on this 
surface. Then \js is a function of the position of P, and by displacing P in any direction 
through a small distance 8s, we find that the flux across the element 8s is given by 
dyjs/ds . 8s. The velocity perpendicular to this element will be 8^jh8s, where h is the thick- 
ness of the stratum, not assumed as yet to be uniform. 

If, further, the motion be irrotational, we shall have in addition a velocity -potential 
0, and the equipotential curves = const, will cut the stream-lines x// = const, at right 
angles. 

In the case of uniform thickness, to which we now proceed, it is convenient to write 
yjs for y\rjh, so that the velocity perpendicular to an element 8s is now given indifferently 
by d\]//ds and dcfy/dn, 8n being an element drawn at right angles to 8s in the proper direction. 
The further relations are then exactly as in the plane problem ; in particular the curves 
= const., \f/ = const., drawn for a series of values in arithmetic progression, the common 
difference being infinitely small and the same in each case, will divide the surface into 
elementary squares. For, by the orthogonal property, the elementary spaces in question 
are rectangles, and if 8s lf 8s 2 be elements of a stream-line and an equipotential line, 
respectively, forming the sides of one of these rectangles, we have d^/ds 2 = d(f>/ds 1 , whence 
8s l = 8s 2 , since by construction 8\f/ = 8<p. 

Any problem of irrotational motion in a curved stratum (of uniform thickness) is 
therefore reduced by orthomorphic projection to the corresponding problem in piano. 
Thus for a spherical surface we may use, among an infinity of other methods, that of 
stereographic projection. As a simple example of this, we may take the case of a stratum 

* 11. c. ante pp. 75, 94. 

t Wiener Sitzungsberichte, lii. 214 (1865) [Wissenschaftliche Abhandlungen, Leipzig, 1909, 
i. 1]. 

J Berl. Monatsber. July 19, 1875 [Ges. Abh. i. 56]. 
§ Pogg. Ann. clx. 375 (1877). 



79-8o] Flow in a Curved Stratum 109 

of uniform depth covering the surface of a sphere with the exception of two circular 
islands (which may be of any size and in any relative position). It is evident that the 
only (two-dimensional) irrotational motion which can take place in the doubly-connected 
space occupied by the fluid is one in which the fluid circulates in opposite directions round 
the two islands, the cyclic constants being equal in magnitude. Since circles project 
into circles, the plane problem is that solved in Art. 64, 2°, viz. the stream-lines are a 
system of coaxal circles with real 'limiting points' (A, B, say), and the equipotential lines 
are the orthogonal system passing through A, B. Returning to the sphere-, it follows from 
well-known theorems of stereographic projection that the stream-lines (including the 
contours of the two islands) are the circles in which the surface is cut by a system of 
planes passing through a fixed line, viz. the intersection of the tangent planes at the 
points corresponding to A and B, whilst the equipotential lines are the circles in which 
the sphere is cut by planes passing through these points*. 

In any case of transformation by orthomorphic projection, whether the motion be 
irrotational or not, the velocity (d\j//dn) is transformed in the inverse ratio of a linear 
element, and therefore the kinetic energies of the portions of the fluid occupying corre- 
sponding areas are equal (provided, of course, the density and the thickness be the same). 
In the same way the circulation (jd\J//dn.ds) in any circuit is unaltered by projection. 

* This example was given by Kirchhoff, in the electrical interpretation, the problem considered 
being the distribution of current in a uniform spherical conducting sheet, the electrodes being 
situate at any two points A, B of the surface. 



CHAPTER V 

1RR0TATI0NAL MOTION OF A LIQUID: PROBLEMS IN 
THREE DIMENSIONS 

81. Of the methods available for obtaining solutions of the equation 

V 2 <£ = (1) 

in three dimensions, the most important is that of Spherical Harmonics. 
This is especially suitable when the boundary conditions have relation to 
spherical or nearly spherical surfaces. 

For a full account of this method we must refer to the special treatises*, 
but as the subject is very extensive, and has been treated from different 
points of view, it may be worth while to give a slight sketch, without formal 
proofs, or with mere indications of proofs, of such parts of it as are most 
important for our present purpose. 

It is easily seen that since the operator V 2 is homogeneous with respect 
to x, y, z, the part of <f> which is of any specified algebraic degree must satisfy 
(1) separately. Any such homogeneous solution of (I) is called a 'spherical 
solid harmonic' of the algebraic degree in question. If cf) n be a spherical 
solid harmonic of degree n, then if we write 

<pn = r"S n , (2) 

S n will be a function of the direction (only) in which the point (x, y, z) lies 
with respect to the origin; in other words, a function of the position of the 
point in which the radius vector meets a unit sphere described with the origin 
as centre. It is therefore called a 'spherical surface harmonic' of order n\. 

To any solid harmonic $ n of degree n corresponds another of degree 
— n — 1, obtained by division by r 2n+1 ; i.e. (j> = r~ 2/l_1 </> n is also a solution of 
(1). Thus, corresponding to any spherical surface-harmonic S n , we have the 
two spherical solid harmonics r n S n and r~ n ~ 1 S n . 

82. The most important case is when n is integral, and when the surface - 
harmonic S n is further restricted to be finite over the unit sphere. In the 

* Todhunter, Functions of Laplace, Lame, and Bessel, Cambridge, 1875. Ferrers, Spherical 
Harmonics, Cambridge, 1877. Heine, Handbuch der Kugelfunctionen, 2nd ed., Berlin, 1878. 
Thomson and Tait, Natural Philosophy, 2nd ed., Cambridge, 1879, i. 171-218. Byerly, Fourier's 
Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, Boston, U.S.A. 1893. Whittaker 
and Watson, Modern Analysis, 3rd ed. , Cambridge, 1920. 

For the history of the subject see Todhunter, History of the Theories of Attraction, dx., 
Cambridge, 1873, ii. Also Wangerin, " Theorie d. Kugelfunktionen, u.s.w.," Encycl. d. math. 
Wiss. ii. (1) (1904). 

t The symmetrical treatment of spherical solid harmonics in terms of Cartesian co-ordinates 
was introduced by Clebsch, in a much neglected paper, Crelle, lxi. 195 (1863). It was adopted 
independently by Thomson and Tait as the basis of their exposition. 



81-82] Spherical Harmonics 111 

form in which the theory (for this case) is presented by Thomson and Tait, 
and by Maxwell*, the primary solution of (1) is 

4>-i = A/r (3) 

This represents as we have seen (Art. 56) the velocity potential due to 
a point-source at the origin. Since (1) is still satisfied when <f> is differ- 
entiated with respect to x, y, or z, we derive a solution 



^- A \ l to+ m fr+ n dik - (4) 



dy 

This is the velocity-potential of a double source at the origin, having its axis 
in the direction (/, m, n); see Art. 56 (3). The process can be continued, 
and the general type of spherical solid harmonic obtainable in this way is 

d 11 1 

*-»- 1= ^ dh 1 dh 2 ...dh M r' (5) 

, 9. . a 3 a 

where aT s = ^ +m ^ +,!s fe' 

l s , m s , n s being arbitrary direction-cosines. 

This may be regarded as the velocity-potential of a certain configuration 
of simple sources about the origin, the dimensions of this system being small 
compared with r. To construct this system we premise that from any given 
system of sources we may derive a system of higher order by first displacing 
it through a space ^7* s in the direction (l s , m s , n s ), and then superposing the 
reversed system, supposed displaced from its original position through a space 
\h s in the opposite direction. Thus, beginning with the case of a simple 
source at the origin, a first application of the above process gives us two 
sources 0+, 0_ equidistant from the origin, in opposite directions. The same 
process applied to the system + , 0_ gives us four sources + + , 0_ + , + _, 
0__ at the corners of a parallelogram. The next step gives us eight sources 
at the corners of a parallelepiped, and so on. The velocity-potential, at a 
great distance, due to an arrangement of 2 n sources obtained in this way, will 
be given by (5), if 4nrA = m'hxh^ ... h n , m' being the strength of the original 
source at 0. The formula becomes exact, for all distances r, when 
h 1} h 2 , ... h n are diminished, and ra' increased, indefinitely, but so that A 
is finite. 

The surface-harmonic corresponding to (5) is given by 

^=^" +1 9 wctJ' (6) 

and the complementary solid harmonic by 

cf> n = r"S n = r^cl>_ n _ 1 (7) 

* Electricity and Magnetism, c. ix. 



112 Irrotational Motion of a Liquid [chap, v 

By the method of 'inversion*,' applied to the above configuration of 
sources, it may be shewn that the solid harmonic (7) of positive degree n 
may be regarded as the velocity-potential due to a certain arrangement of 
2 n simple sources at infinity. 

The lines drawn from the origin in the various directions (l s , m S} n 8 ) are 
called the 'axes' of the solid harmonic (5) or (7), and the points in which 
these lines meet the unit sphere are called the 'poles' of the surface-harmonic 
S n . The formula (5) involves 2n + 1 arbitrary constants, viz. the angular 
co-ordinates (two for each) of the n poles, and the factor A, It can be 
shewn that this expression is equivalent to the most general form of 
spherical surface-harmonic which is of integral order n and finite over the 
unit sphere f. 

83. In the original investigation of Laplace f, the equation V 2 = O is 
first expressed in terms of spherical polar co-ordinates, r, 0, co, where 

x = r cos 0, y = r sin cos co, z = r sin sin co. 

The simplest way of effecting the transformation is to apply the theorem of 
Art. 36 (2) to the surface of a volume-element rS0 .rsin 08co . 8r. Thus the 
difference of flux across the two faces perpendicular to r is 

~-( ^- . r80 . r sin 08co\ 8r. 

Similarly for the two faces perpendicular to the meridian (co — const.) we find 

U%e- rsi » es '°- Sr ) Sd > 

and for the two faces perpendicular to a parallel of latitude (0 = const.) 

oco \r sin uoco } 

Hence, by addition, 



*"s(-59*S(*'»*=»3-* « 

This might of course have been derived from Art. 81 (1) by the usual method 
of change of independent variables. 

If we now assume that cj> is homogeneous, of degree n, and put 

we obtain J- ^(sin 6 ^ +-\^^ + n(n + l)S n = 0, (2) 

sin 000 \ 00 } sin 2 dco 2 

which is the general differential equation of spherical surface-harmonics. 

* Explained by Thomson and Tait, Natural Philosophy, Art. 515. 

t Sylvester, Phil. Mag. (5), ii. 291 (1876) [Mathematical Papers, Cambridge, 1904..., iii. 37]. 
J "Theorie de l'attraction des sph^roides et de la figure des planetes," Mem. de VAcad. roy. 
des Sciences, 1782 [Oeuvres Computes, Paris, 1878... , x. 341]; Mecanique Celeste, Livre 2 me , c. ii. 






82-84] Spherical Harmonics 113 

Since the product n (n + 1) is unchanged in value when we write — n — 1 for 
n, it appears that 

will also be a solution of (1), as already stated (Art. 81). 

84. In the case of symmetry about the axis of x, the term d 2 S n /dco 2 dis- 
appears, and putting cos 6 = fi we get 

l^-^fH^ 1 ^ ' w 

the differential equation of spherical 'zonal' harmonics*. This equation, con- 
taining only terms of two different dimensions in /x, is adapted for integration 
by series. We thus obtain 

( n(n+l) (n-2)n(n+l)(n + 3) 
&„ = ^ljl- 12 /*+ ! 2.3.1 ** "- 

j (n-l)(n + 2) , , (n-3)(«-l)(n + 2)(n+ i» 6 j 

+ B T 1.2.3 ^ + 1.2.3.4.5 -/*--}■ 

(2) 

The series which here present themselves are of the kind called 'hyper- 
geometric'; viz. if we write, after Gauss f, 

F(a, 8,y,x) = l + -=— - a? + -, — z — tf 2 

v ^ /y 1.7 I.2.7.7 + I 

c.g + l.g + 2.fl.£ + l.fl + 2 

+ I.2.3.7.7+I.7 + 2 * ■'•' 

(3) 

we have 

S„ = 41* (- \n, i + £«, i & + B/J.F (| - £», 1 + £w, f , ?) (4) 

The series (3) is of course essentially convergent when x lies between and 1 ; but 
when x=l it is convergent if, and only if, 

y-a-/3>0. 

In this case we have F(a, ft y, l). gkzlll°^|^ ), (5) 

where n(m) is in Gauss's notation the equivalent of Euler's r(ra + l). 
The degree of divergence of the series (3) when 

y-a-/3<0, 
as # approaches the value 1 , is given by the theorem J 

F(a, ft y, #)=(l-#)Y-*-e^( 7 -a, y-ft y, *) (6) 

Since the latter series will now be convergent when x= 1, we see that F(a, /3, y, #) becomes 
divergent as (1 -x)y~ a ~& ; more precisely, for values of x infinitely nearly equal to unity, 
we have 

'^ ^SJttizt^ *-'' " 

ultimately. 

* So called by Thomson and Tait, because the nodal lines (S n = 0) divide the unit sphere into 
parallel belts. 

t I.e. ante p. 104. 

X Forsyth, Differential Equations, 3rd ed., London, 1903, c. vi. 



114 Irrotational Motion of a Liquid [chap, v 



For the critical case where y — a - /3 = 0, 

we may have recourse to the formula 

d Wt- a .. „N_^ 

7 
which, with (6), gives in the case supposed 

7 



Tx F{a, ft y, #)=-f F(a + l, + 1, y + 1, x\ (8) 



^F (a, fr y, x) = ^{\ -x)~K F (y -a, y - $, y+l, x) 



= ^(l-^)-i.^(a,fta+3 + l,^) (9) 

The last factor is now convergent when x = \, so that F (a, ft y, x) is ultimately divergent 
as log (l-x). More precisely we have, for values of x near this limit, 

F ^e>" + ^= n0r&h ) ] <>sih < 10 > 

85. Of the two series which occur in the general expression (Art. 84 (2)) 
of a zonal harmonic, the former terminates when n is an even, and the latter 
when n is an odd integer. For other values of n both series are essentially 
convergent for values of /jl between + 1, but since in each case we have 
y — a — /3 = 0, they diverge at the limits //, = + 1, becoming infinite as 
logO-V). 

It follows that the terminating series corresponding to integral values of 
n are the onhy zonal surface -harmonics which are finite over the unit sphere. 
If we reverse the series we find that both these cases (n even, and n odd) are 
included in the formula* 

_ 1.3.5...(2n-l) | _ n(n-l) 
rnW ~ 1.2. 3. ..7i r 2(2n-l)^ 

n{ n-l)(n-2)(n-S) ) 

+ 2.4,(Z*-l)(2n-3) ^ ""]' V; 

where the constant factor has been adjusted so as to make P n (/x) = l for 
/u, = lf. The formula may also be written 

^to-vhrS^- 1 * (2) 

The series (1) may otherwise be obtained by development of Art. 82 (6), 
which in the case of the zonal harmonic assumes the form 

d n 1 

S n =Ar^^ n - (3) 

dx n r 

* For n even this corresponds to A = ( - Y n ' — r^ , B = ; whilst for n odd we have 

v ; 2 . 4 ... n 

A = 0, B = (-f n - 1] ~ 8 ; 5 V n it • See Heine, i. 12, 147. 
2 . 4 ... (n- 1) 

t Tables of P 1? P 2 , ... P 7 were calculated by Glaisher, for values of fi at intervals of -01, 

Brit. Ass. Report, 1879, and are reprinted by Dale, Five-Figure Tables..., London, 1903. A table 

of the same functions for every degree of the quadrant, calculated under the direction of Prof. 

Perry, was published in the Phil. Mag. for Dec. 1891. Both tables are reproduced in Byerly's 

treatise, also by Jahnke and Emde, Funktionentafeln, Leipzig, 1909. The values of the first 20 

zonal harmonics, at intervals of 5°, have been calculated by Prof. A. Lodge, Phil. Trans. A. cciii. 

(1904). 



84-85] Zonal Harmonics 115 

As particular cases of (2) we have 

PeO*) = l, Px 0*) = /*, P 2 W = i(3/i 2 -l), P 3 (/,) = ! (5/r> -3/4 

Expansions of P n in terms of other functions of 6 as independent variables, 
in places of fx, have been obtained by various writers. For example, we have 

PJ C os^) = l-^^sin^ + (W - 1)n S: + 22 1)(W + 2) sin ^- < 4 > 

This may be deduced from (2)*, or it may be obtained independently by 

putting /jl = 1 — 2z in Art. 84 (1), and integrating by a series. 

The function P n (p) was first introduced into analysis by Legendre t as the coefficient 
of k n in the expansion of 

The connection of this with our present point of view is that if cf> be the velocity-potential 
of a unit source on the axis of x at a distance c from the origin, we have, on Legendre's 
definition, for values of r less than c, 

47r<£ = (c 2 - 2/icr + r 2 )~h 

T l* 2 

-£+*?+*?+ < B > 

Each term in this expansion must separately satisfy V 2 (£ = 0, and therefore the coefficient 
P n must be a solution of Art. 84 (1). Since P n , as thus defined, is obviously finite for all 
values of /x, and becomes equal to unity for /x= 1, it must be identical with (1). 

For values of r greater than c, the corresponding expansion is 

^=l+ p 4 +p 4 + (6) 

We can hence deduce expressions, which will be useful to us later, Art. 98, for the 
velocity-potential due to a double-source of unit strength, situate on the axis of x at a 
distance c from the origin, and having its axis pointing from the origin. This is evidently 
equal to dcf)/dc, where $ has either of the above forms; so that the required potential is, 
for r<c, 

-l&* p *? +3P 4- ■■■) < 7 > 

andforr> c , s{ Pl h +iP ^*-) (8) 

The remaining solution of Art. 84 (1), in the case of n integral, can be 
put into the more compact form} 

e»o*)=iP»(^)iog^-^, (9) 

where *_-_. P^ + — ^ P„_, + (10) 

* Murphy, Elementary Principles of the Theories of Electricity, <£c, Cambridge, 1833, p. 7. 
[Thomson and Tait, Art. 782.] 

t "Sur l'attraction des spheroides homogenes," Mem, des Savans Etrangers, x. (1785). 

% This is equivalent to Art. 84 (4) with, for n even, A = 0, B=(-)$ n : — m " n ; whilst for 



n odd we have^ = (-)* (w+1) 2 - a 4 -:- (n X) , B = 0. See Heine, i. 141, 147. 

5 . o ... n 



116 Irrotational Motion of a Liquid [chap, v 

This function Q n ^) is sometimes called the zonal harmonic 'of the second 
kind.' 

Thus 
Qo M - i log \±£, Q, W = 1 (3/, 2 - 1) log \±t - ifli 

A. — fJL JL — fJU 

86. When we abandon the restriction as to symmetry about the axis of x, 
we may suppose S n , if a finite and single- valued function of o>, to be expanded 
in a series of terms varying as cos sw and sin sco respectively. If this expansion 
is to apply to the whole sphere (i.e. from co = to co = 2tt), we may further (by 
Fourier's theorem) suppose the values of s to be integral. The differential 
equation satisfied by any such term is 

|^-^f} + H» +i )-^} s »= o « 

If we put S n = (1 - fi 2 )^ s v, 

this takes the form 

(i_ ' t2) S" 2(s+i)M £ +( "" s)(n+s+i)t,=0, 

which is suitable for integration by series. We thus obtain 

t (n - s - 2) (n - s) (n + s + 1) (n -f s + 3) 4 ) 

+ 1.2.3.4 A*--v| 

, (n - g - 3) (n - j - l)(n + g + 2) (n + j + 4) " B ) ,~ 



1.2.3.4.5 

the factor cos sco or sin s&> being for the moment omitted. In the hyper- 
geometric notation this may be written 

Sn-il-^lAF^s-in^ + is + in,^^) 

+ BfMF(i + is-in,l+liS + in,§,^)}. ...(3) 

These expressions converge when /a 2 < 1, but since in each case we have 

y — a — /9 = -5, 

the series become infinite as (1 - /j?)~ s at the limits /n = + 1, unless they 
terminate*. The former series terminates when n — s is an even, and the 

* Kayleigh, Theory of Sound, London, 1877, Art. 338. 



85-86] Tesseral Harmonics 117 

latter when it is an odd integer. By reversing the series we can express both 
these finite solutions by the single formula* 

P s (a) _ ( 2n ) 1 a _ ^h Ln- S _ {n-s){n-s-l) 2 

rn W -2"(n-s)\n\ {i M r 2.(2n-l) ^ 

(ft-s)(n-s-l)(ft-s-2)(?i-s-3) _ 4 _ ) , . 
"*"" 2.4.(2n-l)(2n-3) ^ — J-— v ; 

On comparison with Art. 85 (1) we find that 

P n 'W = (l-^*™> (5) 

That this is a solution of (1) may of course be verified independently. 

In terms of sin \ 0, we have 

d «/ a\ (n+s)\ • ,/jf-, (n-s)(n + s + l) . 21z) 

P n s (cos 0) = —, f— — sm s ^ 1 - =-^ z-r sin 2 \Q 

71 v ' 2*(n-s)\s\ \ 1 .(s + 1) 2 

+ 1.2.(. + l)(. + 2) sin^-...|....(6) 

This corresponds to Art. 85 (4), from which it can easily be derived. 

Collecting our results we learn that a surface-harmonic which is finite over 
the unit sphere is necessarily of integral order, and is further expressible, if 
n denote the order, in the form 

S n = A P n (ji) + 2, (A 8 cos sm +B 8 sin SG>)P n 8 (ti), (7) 

containing 2n 4- 1 arbitrary constants. The terms of this involving a> are 
called ' tesseral ' harmonics, with the exception of the last two, which are given 
by the formula 

(1 — fj?)% n (A n cos nco + B n sin nay), 

and are called ' sectorial ' harmonics f ; the names being suggested by the forms 
of the compartments into which the unit sphere is divided by the nodal lines 

S n = 0. 

The formula for the tesseral harmonic of rank s may be obtained otherwise 
from the general expression (6) of Art. 82 by making n — s out of the n poles 
of the harmonic coincide at the point 6 = of the sphere, and distributing the 
remaining s poles evenly round the equatorial circle 6 = \tt. 

The remaining solution of (1), in the case of n integral, may be put in 
the form 

S n = ( A 8 cos sco + B s sinsco) Q n s (^), (8) 

* There are great varieties of notation in connection with these 'associated functions,' as 
they have been called. That chosen in the text was proposed by F. Neumann ; and is adopted by 
Whittaker and Watson, p. 323. 

f The prefix ' spherical ' is implied ; it is often omitted for brevity. 



118 Irrotational Motion of a Liquid [chap, v 

where* q^^.^*^ (9 ) 

This is sometimes called a tesseral harmonic ' of the second kind.' 

87. Two surface-harmonics S, S' are said to be ' conjugate/ or ; orthogonal/ 
when 

ffSS'dm = 0, (1) 

where 8vr is an element of surface of the unit sphere, and the integration ex- 
tends over this sphere. 

It may be shewn that any two surface-harmonics, of different orders, 
which are finite over the unit sphere, are orthogonal, and also that the 2n + 1 
harmonics of any given order n, of the zonal, tesseral, and sectorial types 
specified in Arts. 85, 86, are all mutually orthogonal. It will appear, later, 
that the orthogonal property is of great importance in the physical applications 
of the subject. 

Since 8vr = sin 6808(o = — 8fju8co, we have, as particular cases of this 
theorem, 

J 1 i P m ( / ,)^ = 0, (2) 

J* P.W.P.00<*/**0, (3) 

J 1 P m s (ri.P n s(fjL)dp = 0, (4) 



and 



provided m, n are unequal. 

For m — n, it may be shewn f that 



jp^^d^^j, (5) 

>w=^;ra <«> 



88. We may also quote the theorem that any arbitrary function f(fi, co) 
of the position of a point on the unit sphere can be expanded in a series of 
surface-harmonics, obtained by giving n all integral values from to oo , in 
Art. 86 (7). The formulae (5) and (6) are useful in determining the coefficients 
in this expansion. 

Thus, in the case of symmetry about an axis, the theorem takes the form 

/( A *) = Co+CiP 1 ( M ) + CiP 1 0*)+... + (7 n P fl (/*)+ (7) 

If we multiply both sides by P n (/uu) dp, and integrate between the limits ± 1, 
we find 

Ck-if f(r)dn (8) 

* A table of the functions Q n (/a), Q n 8 (/x), for various values of n and s, is given by Bryan, Proc. 
Camb. Phil. Soc. vi. 297. 

t Ferrers, p. 86 ; Whittaker and Watson, pp. 306, 325. 



86-89] Integral Formulae 119 

and, generally, 

C.-^£±\_ i f(t*)P.(M)dr (9) 

For the analytical proof of the theorem recourse must be had to the 
special treatises*; the physical grounds for assuming the possibility of this 
and other similar expansions will appear, incidentally, in connection with 
various problems. 

89. Solutions of the equation V 2 <f> = may also be obtained by the usual 
method of treating linear equations with constant coefficients f. Thus, the 
equation is satisfied by 

or, more generally, by <f> = f(ax + fty 4- yz), (1) 

provided a 2 + /3 2 + 7 2 = (2) 

For example, we may put 

ff,ft7=l, icosS-, isin^r, (3) 

or, again, a, /3, 7 = 1, icoshu, sinh u (4) 

It may be shewn % that the most general solution possible can be obtained by 
superposition of solutions of the type (1). 

Using (3), and introducing the cylindrical co-ordinates x, m, &>, where 

2/ = tfrcoso>, 2=Grsin&), (5) 

we build up a solution symmetrical about the axis of x if we take 

1 [ 2ir 

<j) = ^r— f{m + VSF COS (Sr — ©)} d&. 

Zir J o 

For, since the integration extends over a whole circumference, it is immaterial 
where the origin of ^ is placed, and the formula may therefore be written § 

1 [ 2n 1 f* 
<£ = ^- f (a? + im cos %)d& = - f(x + ivrcos%)d% (6) 

&7T J IT J 

This is remarkable as giving a value of <f>, symmetrical about the axis of 
x, in terms of its values f(x) at points of this axis. It may be shewn, by means 
of the theorem of Art. 38, that the form of <£ is in such a case completely 
determined by the values over any finite length of the axis||. 

As particular cases of (6) we have the functions 

- f * (x + %v cos *)* d&, - f " (x + t«r cos ^)- w " 1 d&, 

7TJ IT J o 

* For an account of the more recent investigations of the question, see Wangerin, I.e. 

t Forsyth, Differential Equations, p. 444. 

% Whittaker, Month. Not. B. Ast. Soc. lxii. (1902). 

§ Whittaker and Watson, Modern Analysis, c. xviii. 

II Thomson and Tait, Art. 498. 



120 Irrotational Motion of a Liquid [chap, v 

where n will be supposed to be integral. Since these are solid harmonics finite 
over the unit sphere, and since, for vr = 0, they reduce to r n and r~ n ~\ they 
must be equivalent to P n (/jl) r n , and P n (fj) r _n_1 , respectively. We thus 
obtain the forms 



iY0") = - [*{** + *V(1 -^ 2 )cos^^, (7) 

7T Jo 
P n (?) = - J Q ^ + i ^/ (1 _ ^ cog c^jn+l • ( 8 ) 



due originally to Laplace* and Jacobif, respectively. 

90. As a first application of the foregoing theory let us suppose that an 
arbitrary distribution of impulsive pressure is applied to the surface of a 
spherical mass of fluid initially at rest. This is equivalent to prescribing an 
arbitrary value of <£ over the surface ; the value of </> in the interior is thence 
determinate, by Art. 40. To find it, we may suppose the given surface-value 
to be expanded, in accordance with the theorem quoted in Art. 88, in a series 
of surface-harmonics of integral order, thus 

<l> = So + S 1 + S t + ...+S n + (1) 

The required value is then 

for this satisfies V 2 <£ = 0, and assumes the prescribed form (1) when r— a, the 
radius of the sphere. 

The corresponding solution for the case of a prescribed value of <£ over 
the surface of a spherical cavity in an infinite mass of liquid initially at rest 
is evidently 

a ~ a 2 „ a 3 a n+1 ^ 
<*>—«»+ J3&+ - 3 « 2 + ... + ^s+i«»+ (3) 

Combining these two results we get the case of an infinite mass of fluid 
whose continuity is interrupted by an infinitely thin vacuous stratum, of 
spherical form, within which an arbitrary impulsive pressure is applied. The 
values (2) and (3) of <f> are of course continuous at the stratum, but the 
values of the normal velocity are discontinuous, viz. we have, for the internal 
fluid, 



or a 



and for the external fluid 



§* __:£(„ + 1)* 

or 'a 

* Mec. Cel. Livre ll me , c. ii. 

t Crelle, xxvi. (1843) [Gesammelte Werke, Berlin, 1881... , vi. 148]. 



89-9i] Applications 121 

The motion, whether internal or external, is therefore that due to a 
distribution of simple sources with surface-density 



2(2n+l)^ (4) 



over the sphere ; see Art. 58. 



91. Let us next suppose that, instead of the impulsive pressure, it is the 
normal velocity which is prescribed over the spherical surface ; thus 

d £ = S 1 + S i +...+S n +..., (1) 

the term of zero order being necessarily absent, since we must have 

d<f> 



w 



,,.^ = 0, (2) 



on account of the constancy of volume of the included mass. 

The value of <f> for the internal space is of the form 

<f> = A 1 rS 1 + A 2 r 2 S 2 +...+A n r n S n + ..., (3) 

for this is finite and continuous, and satisfies V 2 <£ = 0, and the constants can 
be determined so as to make d<f>/dr assume the given surface- value (1); viz. 
we have 7iJ. n a n_1 = 1. The required solution is therefore 

1 r n 
<t> = a2±^-S n (4) 

Y na n n v ' 

The corresponding solution for the external space is found in like manner 
to be 

*=- a% l^% 8 » (5) 

The two solutions, taken together, give the motion produced in an infinite 
mass of liquid which is divided into two portions by a thin spherical membrane, 
when a prescribed normal velocity is given to every point of the membrane, 
subject to the condition (2). 

The value of <f> changes from aXS n /n to — aXS n /(n + 1), as we cross the 
membrane, so that the tangential velocity is now discontinuous. The motion, 
whether inside or outside, is that due to a double-sheet of density 

-» 2 ^^ w 

see Art. 58. 

The kinetic energy of the internal fluid is given by the formula (4) of 
Art. 44, viz. 

2T=pjJ4> d ^dS^pa^ljjS^d^ (7) 

the parts of the integral which involve products of surface-harmonics of 
different orders disappearing in virtue of the orthogonal property of Art. 87. 



122 Irrotational Motion of a Liquid [chap, v 

For the external fluid we have 

*T — p\\*%iS-par2.^ rx \\lUdm (8) 

91 a. The harmonic of zero order lends itself at once to the discussion of 
the two mathematically cognate problems of the collapse of a spherical bubble 
yi water, and the expansion of a spherical cavity due to the pressure of an 
included gas, as in the case of a submarine mine. 

In the former problem*, if R Q be the initial radius of the bubble, and R its value at 
time t, we have 

*-*?, a) 

since this makes — dcp/dr=R, for r=R. Hence, putting G = in Art. 22 (5), we have 

p-po _ R 2 R + 2RR* _ RtR 2 m 

p ~ r 2r* ' { } 

if p be the pressure at r=oo . Hence, putting r — R and neglecting the internal pressure 

RR+%&=-?-\ (3) 

the integral of which is jfil&=%& (Ro 3 -R 3 ) (4) 

This cannot easily be integrated further, but the time fa) of total collapse can be found ; 
thus, putting R = R x^ y 

^^N/(4)/o*"* (1 -* r4 ^=^V(^ I ^ )= " 916 * ,V0>w --" t5) 

Thus if p = l, Ro=I cm., and ^ =l° 6 C.G.s. (1 atmosphere), ^ = -000915 sec. 

The kinetic energy at any instant is 

27rpR?R 2 = §np (R 3-R 3 ), (6) 

as is indeed obvious from a consideration of the work done at a distance on the fluid. 
When, the collapse occurs, the energy destroyed, or rather converted into other forms, is 
inpoRo 3 . If ^0 = 1, Po= 10 6 , this is 4'18 x 10 6 ergs, or about -308 of a ft. -lb. 

The equations (1) and (2) are applicable also to the problem of the expanding cavity, 
but we now negleut the pressure p at a distance. If p t be the initial pressure in the cavity, 
when R = R , and i2=0, the internal pressure at time t is given by 

,H§f <* 

if we assume the adiabatic law of expansion. Hence 

RR+%R2 = c *(^\ (8) 

where c = \/(M>) (9) 

This quantity c is of the nature of a velocity, and determines the rapidity with which 
changes take place. The integral of (8) is 

$-,-^m-(m <■»> 

* Besant, Hydrostatics and Hydrodynamics, Cambridge, 1859; Kayleigh, Phil. Mag. xxxiv. 
94 (1917) [Papers, vi. 504]. 



91-92] Radial Motion 123 

It appears from (8) that the initial acceleration (R) in the radius is c 2 /i2 , whatever the 
law of expansion. For (8) and (10) we find that the maximum of i? occurs when 

(RJR fy-* = y, (11) 

and is given by — s = ; — 77 (12) 

The solution is not easily completed except in the special case of y = $. Writing 

R/fio=l+z, (13) 

we have then (i +2 )2^ = ^ // 2 A (14) 

■ - 
whence c */22 = > /(2;s)(l+§z+^ 2 ). (15) 

As a concrete illustration, suppose the initial diameter of the cavity to be 1 metre, and 
the initial pressure pi to be 1000 atmospheres, which makes c = 3'16x 10 4 cm. /sec. It is 
then found that the radius of the cavity is doubled in ^\q of a second, and multiplied 
five-fold in about -fa sec. The initial acceleration of the radius is 2'OOx 10 7 cm./sec. 2 , 
shewing that the neglect of gravity in the early stages of the motion is amply justified. 
The maximum of R occurs when R/R = $, £='0016 sec, and is about 145 metres per 
second, or about one-tenth of the velocity of sound in water. With initial pressures of the 
order of 10,000 atmospheres or more, we should have velocities comparable with the velocity 
of sound, and the effect of compressibility would be no longer negligible*. 

92. The harmonic of the first order is involved in the problem of the 
motion of a solid sphere in an infinite mass of liquid which is at rest at infinity. 
If we take the origin at the centre of the sphere, and the axis of x in the 
direction of motion, the normal velocity at the surface is Ux/r, = £7" cos 0, where 
U is the velocity of the centre. Hence the conditions to determine <£ are (1°) 
that we must have V 2 <£ = everywhere, (2°) that the space-derivatives of <t> 
must vanish at infinity, and (3°) that at the surface of the sphere (r = a) we 
must have 

- d ^=Ucosd (1) 

or 

The form of this suggests at once the zonal harmonic of the first order; we 
therefore assume 

,81 . cos 6 

ox r r 1 

The condition (1) gives — 2A/a?= U, so that the required solution isf 

= £ET^cos0 (2) 

It appears on comparison with Art. 56 (4) that the motion of the fluid is 
the same as would be produced by a double-source of strength 27r Ua?, situate 
at the centre of the sphere. For the forms of the lines of motion see p. 128. 

* This discussion is taken from a paper "The early stages of a submarine explosion," Phil. 
Mag. xlv. 257 (1923). 

t Stokes, "On some cases of Fluid Motion," Camb. Trans, viii. (1843) [Paper?, i. 17]. Dirichlet, 
"Ueber die Bewegung eines festen Korpers in einem incompressibeln fliissigen Medium," Berl. 
Monatsber. 1852 [Werke, Berlin, 1889-97, ii. 115]. 



124 Irrotational Motion of a Liquid [chap, v 

To find the energy of the fluid motion we have 

2T=- P (U d ^dS = $ P aU 2 rcos 2 d.27rasm0.add 

= f7r /3 a 3 /7 2 = i/ / ^ 2 , (3) 

if M' — \ irpa z . It appears, exactly as in Art. 68, that the effect of the fluid 
pressure is equivalent simply to an addition to the inertia of the solid, the 
increment being now half the mass of the fluid displaced *. 

Thus in the case of rectilinear motion of the sphere, if no external forces 
act on the fluid, the resultant pressure is equivalent to a force 

- M,d £> <*> 

in the direction of motion, vanishing when U is constant. Hence if the sphere 
be set in motion and left to itself, it will continue to move in a straight line 
with constant velocity. 

The behaviour of a solid projected in an actual fluid is of course quite 
different; a continual application of force is necessary to maintain the motion, 
and if this be not supplied the solid is gradually brought to rest. It must be 
remembered, however, in making this comparison, that in a 'perfect' fluid 
there is no dissipation of energy, and that if, further, the fluid be incompressible, 
the solid cannot lose its kinetic energy by transfer to the fluid, since, as we 
have seen in Chapter ill., the motion of the fluid is entirely determined by 
that of the solid, and therefore ceases with it. 

If we wish to verify the preceding result by direct calculation from the formula 

;-¥-**+*(* m 

we must remember that the origin is in motion, and that the values of r and 6 for a fixed 
point of space are therefore increasing at the rates — £7 cos 0, and (ZJsin 6)/r, respectively ; 
or we may appeal to Art. 20 (6). In either way we find 

P = $a^COs6 + 1 \U2cOS20-^U* + F(t) ( 6 ) 

The last three terms are the same for surface-elements in the positions Q and it — 6 ; so 
that, when U is constant, the pressures on the various elements of the anterior half of the 
sphere are balanced by equal pressures on the corresponding elements of the posterior 
half. But when the motion of the sphere is being accelerated there is an excess of pressure 
on the anterior, and a defect on the posterior half. The reverse holds when the motion is 
being retarded. The resultant effect in the direction of motion is 

dU 



as before. 



■/, 



27ra sin 6 . ad6 . p cos &= — %7rpa 3 _ 



93. The same method can be applied to find the motion produced in a 
liquid contained between a solid sphere and a fixed concentric spherical 
boundary, when the sphere is moving with given velocity U. 

* Stokes, I.e. The result had been obtained otherwise, on the hypothesis of infinitely small 
motion, by Green, "On the Vibration of Pendulums in Fluid Media," Edin. Trans. 1833 [Papers, 
p. 315]. 



92-94] Motion of a Sphere 125 

The centre of the sphere being taken as origin, it is evident, since the space occupied 
by the fluid is limited both externally and internally, that solid harmonics of both positive 
and negative degrees are admissible ; they are in fact required, in order to satisfy the 

boundary conditions, which are 

— 9<£/9r = UcosB, 

for r = a, the radius of the spheres, and 

for r=b, the radius of the external boundary, the axis of x being as before in the direction 
of motion. 

We therefore assume <£= \Ar + -\ cos B, (1) 

and the conditions in question give 

. 2B TT . 2B 

a 3 b 3 ' 

whence A= V ^-,U, B=ij^- 3 U. ,..(2) 

b 3 - a 3 *b 3 — a 3 

The kinetic energy of the fluid motion is given by 

the integration extending over the inner spherical surface, since at the outer we have 
d<j>/dr=0. We thus find 

2 ^=i"^^V^ 2 ( 3 ) 

It appears that the effective addition to the inertia of the sphere is now* 

2 & 3 + 2« 3 3 fA , 

^ w^ pa (4) 

As b diminishes from oo to a, this increases continually from %irpa 3 to oo , in accordance 
with Lord Kelvin's minimum theorem (Art. 45). In other words, the introduction of a 
rigid spherical partition in the problem of Art. 92 acts as a constraint increasing the 
kinetic energy for any given velocity of the sphere, and so virtually increasing the inertia 
of the system. 

94. In all cases where the motion of a liquid takes place in a series of 
planes passing through a common line, and is the same in each such plane, 
there exists a stream-function analogous in some of its properties to the two- 
dimensional stream-function of the last Chapter. If in any plane through the 
axis of symmetry we take two points A and P, of which A is arbitrary, but 
fixed, while P is variable, then considering the annular surface generated by 
any line AP, it is plain that the flux across this surface is a function of the 
position of P. Denoting this function by 27r\/r, and taking the axis of x to 
coincide with that of symmetry, we may say that ^isa function of x and -cr, 
where x is the abscissa of P, and <bt, = (y 2 4- z 2 )^, is its distance from the axis. 
The curves yjr = const, are evidently stream-lines. 

If P' be a point infinitely near to P in a meridian plane, it follows from 
the above definition that the velocity normal to PP' is equal to 

27T^.PP" 

* Stokes, I.e. ante p. 123. 



126 Irrotational Motion of a Liquid [chap, v 

whence, taking PP' parallel first to tss and then to x, 

u = --^-, v = -^, (1) 

•us ors ns ox 

where u and v are the components of fluid velocity in the directions of x and 
ns respectively, the convention as to sign being similar to that of Art. 59. 

These kinematical relations may also be inferred from the form which the 
equation of continuity takes under the present circumstances. If we express 
that the total flux into the annular space generated by the revolution of an 
elementary rectangle SxBnr is zero, we find 

?r- (U . 27TOT Sct) Sx + ^— (v . 277-1*7 Bx) 8vS = 0, 

ox x Ovs 

or S (fn * ) + ^ (iro) " 0i (2) 

which shews that wv.dx-7jsu.d7s 

is an exact differential. Denoting this by dty we obtain the relations (1)*. 

So far the motion has not been assumed to be irrotational; the condition 
that it should be so is 

dv du _ 

OX Ons ' 

which leads to g + J±_I|± = o (3) 

Oar 0ns 2 ns Ons 

The differential equation of <f> is obtained by writing 

d<l> dcf> 

ox 0ns 

in (2), viz. it is -X + ^ + _-^l = o (4) 

Ox 2 dm 2 nsdns 

It appears that the functions <£ and yjr are not now (as they were in Art. 62) 

interchangeable. They are, indeed, of different dimensions. 

The kinetic energy of the liquid contained in any region bounded by 

surfaces of revolution about the axis is given by 

: ,;= lU^dS 



■~'K 

= 2irp [</>cty, (5) 



— ¥■. 2irnsds 
nsOs 



The stream-function for the case of symmetry about an axis was introduced in this manner 
by Stokes, "On the Steady Motion of Incompressible Fluids," Camb. Trans, vii. (1842) [Papers, 
i. 1]. Its analytical theory has been treated very fully by Sampson, "On Stokes' Current- 
Function," Phil. Trans. A, clxxxii. (1891). 



94-95] Stokes' Stream-Function 127 

8s denoting an element of the meridian section of the bounding surfaces, and 
the integration extending round the various parts of this section, in the proper 
directions. Compare Art. 61 (2). 

95. In the case of a point-source at the origin whose velocity-potential is 

4>=l (i) 

the flux through any closed curve is numerically equal to the solid angle 
which the curve subtends at the origin. Hence for a circle with Ox as axis, 
whose radius subtends an angle 6 at 0, we have, attending to the sign, 

2tt^ = - 2tt (1 - cos d). 
Omitting the constant term we have 

+-;-£ • ■••*■> 

The solutions corresponding to any number of simple sources situate at 

various points of the axis of x may evidently be superposed; thus for the 

double-source 

a 1 cos . . 

* = -^r = ^' (3) 

■ r Wr v* sin 2 B ... 

wehave ^ = -^ = "^ = "T _ (4) 

And, generally, to the zonal solid harmonic of degree — n — 1, viz. to 

♦- k £i (5) 

corresponds * ty — A 5-^73 (6) 

A more general formula, applicable to harmonics of any degree, fractional 
or not, may be obtained as follows. Using spherical polar co-ordinates r, 0, 
the component velocities along r, and perpendicular to r in the plane of the 
meridian, are found by making the linear element PP' of Art. 94 coincide 
successively with rh6 and Br, respectively, viz. they are 

r sin 6 rd6 y rsind dr 

Hence in the case of irrotational motion we have 

JjL-,*t*, ? — «ntf|[ (8) 

sin Odd dr dr W 

Thus if $ = r n S ni (9) 

where S n is a zonal harmonic of order n, we have, putting p = cos 6, 

* Stefan, "Ueber die Kraftlinien eines um eine Axe symmetrischen Feldes," Wied. Ann. 
xvii. (1882). 



128 Irrotational Motion of a Liquid [chap, v 

The latter equation gives 

*=;rhr n+1(1 -" 2) f' -m 

which must necessarily also satisfy the former; this is readily verified by 

means of Art. 84 (1). 

Thus in the case of the zonal harmonic P n , we have as corresponding values 

1 dP 
* = *-P„<aO, + __£- f -H»(l_ # *>5g (11) 

1 dP 

and </ > = r- w - 1 P n ( M ), ^ = - - r~ n (I - p*) "^ , (12) 

Tl CifjL 

of which the latter must be equivalent to (5) and (6). The same relations 
hold of course with regard to the zonal harmonic of the second kind, Q n . 

96. We saw in Art. 92 that the motion produced by a solid sphere in 
an infinite mass of liquid may be regarded as due to a double-source at the 




95-96] Stream-Lines of a Sphere 129 

centre. Comparing the formulae there given with Art. 95 (4), it appears that 
the stream-function due to the sphere is 

^ = -itf-sin 2 (1) 

The forms of the lines of motion corresponding to a number of equidistant 
values of yjr are shewn on the opposite page. The stream-lines relative to the 
sphere are figured in a diagram near the end of Chapter VII. 

Again, the stream-function due to two double-sources having their axes oppositely 
directed along the axis of x will be of the form 

*-7?— 7?-' (2) 

where r ly r 2 denote the distances of any point from the positions P and Q, say, of the two 
sources. At the stream-surface | = Owe have 

ri /r 2 = (A/B)K 

i.e. the surface is a sphere in relation to which P and Q are inverse points. If be the 
centre of this sphere, and a its radius, we find 

AIB=OP s la 3 =a*IOQ 3 (3) 

This sphere may be taken as a fixed boundary to the fluid on either side, and we thus 
obtain the motion due to a double-source (or say to an infinitely small sphere moving 
along Ox) in presence of a fixed spherical boundary. The disturbance of the stream-lines 
by the fixed sphere is that due to a double-source of the opposite sign placed at the 
'inverse' point, the ratio of the strengths being given by (3)*. This fictitious double- 
source may be called the ' image ' of the original one. 

There is also a simple construction for the image of a point-source in a fixed sphere. 
The image of a source m at P will consist of a source m . OQ/a at the inverse point Q, 
together with a line of sinks extending with uniform line-density - mja from P to the 
centre Of. 

This might be deduced by integration from the preceding result, but a direct verifica- 
tion is simpler. It follows at once from Art. 95 (2) that the stream-function due to a line 
of sources of density m would be 

y\r = m{r- r'), (4) 

where r, r are the distances of the two ends of the line from the point considered. Hence 
the arrangement of sources just described will give, at any point R on the sphere, 

^= -m. cos RPO-m. ^§cos 0QR-- (OR-QR) (5) 

Since QR = OR cos ORQ + OQ cos OQR, and RPO = ORQ, 

this reduces to \jr = — m, a constant over the sphere. 

For the calculation of the force on the sphere we have recourse to zonal harmonics. 
Referred to as origin the velocity-potential of the original source, in the neighbourhood 
of the sphere, is given by 

,. 1 rcosB r 2 (3cos 2 0-l) 

* This result was given by Stokes, " On the Resistance of a Fluid to two Oscillating Spheres," 
Brit. Ass. Report, 1847 [Papers, i. 230]. 

t Hicks, I.e. infra, p. 134. See the diagram on p. 71 ante. 



130 Irrotational Motion of a Liquid [chap, v 

The motion reflected from the sphere will be given by 

a 3 cos0 a 6 (3cos 2 0-l) 

»/»— W+ 3cM + "> (7) 

since this makes djdr (<f> + <f>') = 0, for r = a. The velocity at the surface will therefore be 
d . , ,, v 3m . „ 5ma . „ 

<?= "^ (< ^" f0)== 2^ Sm ^ + ^3- Sm ^ COS ^ + (8) 

For an approximate result we may stop the expansion at this point. The resultant 
force towards P is then 

X=-f n pcosd.27ra 2 smede = 7r P a 2 f V q 2 sm6cosede = ^^- (9) 

If/ be the acceleration at when the sphere is absent, viz. f=2m 2 /c 5 , we have 

Z = 27rpa 3 /. (10)*. 

97. Rankinef employed a method similar to that of Art. 71 to discover 
forms of solids of revolution which will by motion parallel to their axes 
generate in a surrounding liquid any given type of irrotational motion 
symmetrical about an axis. 

The velocity of the solid being U, and Bs denoting an element of the 
meridian, the normal velocity at any point of the surface is Udtxr/ds, and that 
of the fluid in contact is given by — d^lixds. Equating these and integrating 
along the meridian, we have 

^ = -|^ 2 + const (1) 

If in this we substitute the value of yfr due to any distribution of sources 
along the axis of symmetry, we obtain the equation of a family of stream- 
lines. If the sum of the strengths is zero, one of these lines will serve as the 
profile of a finite solid of revolution past which the flow takes place. 

In this way we may readily verify the solution already obtained for the 
sphere; thus, assuming 

T/r = ^*7 2 /r 3 , (2) 

we find that (1) is satisfied for r — a, provided 

A = -iUa? } (3) 

which agrees with Art. 96 (1). 

By a continuous distribution of sources and sinks along the axis it has 
been found possible to imitate forms which have empirically been found 
advantageous for the profiles of air-ships. The fluid pressures can in such 
cases be calculated, and the results compared with experiment. 

98. The motion of a liquid bounded by two spherical surfaces can be 

found by successive approximations in certain cases. For two solid spheres 

moving in the line of centres the solution is greatly facilitated by the result 

given at the end of Art. 96, as to the 'image' of a double-source in a fixed 

sphere. 

* Prof. G. I. Taylor, Aeronautical Research Committee, R. & M. 1166 (1928). 
f "On the Mathematical Theory of Stream Lines, especially those with Four Foci and 
upwards," Phil. Trans. 1871, p. 267 (not included in the collection referred to on p. 63 ante). 



96-98] 



Motion of Two Spheres 



131 



Let a, b be the radii, and c the distance between the centres A, B. Let U be the 
velocity of A towards B, U' that of B towards A. Also, P being any point, let AP—r, 
BP=r', PAB=6, PBA = 0'. The velocity-potential will be of the form 

U<\>+U'4>', (1) 

where the functions <f> and <£' are to be determined by the conditions that 

V 2 = O, vy=0, (2) 




throughout the fluid, that their space-derivatives vanish at infinity, and that 






or 



over the surface of A. whilst 



30 

a/ 



= 0, 



dr 






.(3) 



cos 6\ 



.(4) 



over the surface of B. It is evident that cf> is the value of the velocity-potential when A 
moves with unit velocity towards B, while B is at rest ; and similarly for <£'. 

To find <p, we remark that if B were absent the motion of the fluid would be that due 
to a certain double-source at A having its axis in the direction AB. The theorem of Art. 96 
shews that we may satisfy the condition of zero normal velocity over the surface of B by 
introducing a double-source, viz. the ' image ' of that at A in the sphere B. This image is 
at ff l3 the inverse point of A with respect to the sphere B; its axis coincides with AB, and 
its strength is -/zo^/c 3 , where fi is the strength of the original source at A, viz. 



Mo 



= 27T« 3 . 



The resultant motion due to the two sources at A and H\ will however violate the condition 
to be satisfied at the surface of the sphere A, and in order to neutralize the normal velocity 
at this surface, due to H u we must superpose a double-source at H^ the image of H x in 
the sphere A. This will introduce a normal velocity at the surface of B, which may again 
be neutralized by adding the image of ff 2 m &i an( ^ so on « If Mi> /*2> M3> ••• De tne strengths 
of the successive images, and/i,/|,/|, ... their distances from A, we have 



/ 3 = c " 



6 2 


/l _ 7l 
/l 


Ml & 3 

MO <* 


M2 a% \ 


6 2 




M3 & 3 


M4 « 3 

M3 /3 3 ' 


c-ft 


M2 (c-/ 2 ) 3 ' 


6 2 


Jb 


M5 & 3 


Me a 3 


c-A' 


M4 (^.A) 3 ' 


M5 /5 3 '/ 






•(5) 



and so on, the laws of formation being obvious. The images continually diminish in 
intensity, and this very rapidly if the radius of either sphere is small compared with the 
shortest distance between the two surfaces. 



132 Irrotational Motion of a Liquid [chap, v 

The formula for the kinetic energy is 

2T=-p[[(U<l>+U'4>') ((7 d ^+U' d ^\dS=LU* + 2MUU' + NU' 2 , (6) 



provided 



a) 



where the suffixes indicate over which sphere the integration is to be effected. The 
equality of the two forms of M follows from Green's Theorem (Art. 44). 

The value of <p near the surface of A can be written down at once from the results (7) 
and (8) of Art. 85, viz. we have 

4^ = ( M0 + /x2 + M4+...) C -^-2(^4-^3-r...)rcos^ + &c., 

the remaining terms, involving zonal harmonics of higher orders, being omitted, as they 
will disappear in the subsequent surface-integration, in virtue of the orthogonal property 
of Art. 87. Hence, putting d<p/dn= — cos 0, we find with the help of (5) 



.(8) 



-£=Jp(/«o + 3M2 + 3/^ + ...)«Swpa»( 1+3^ + 3 



a 6 6 6 



c?fi 3 *ffifi-ftfff 



•) (9) 



It appears that the inertia of the sphere A is in all cases increased by the presence of a 
fixed sphere B. Compare Art. 93. 

The value of N may be written down from symmetry, viz. it is 

1 + 3^ + 3 -_g-g— — + ...), (10) 



where 



1=C " J 



fs=c- 



fl 



&2 ^ 

"fl 



c-/ 2 ' 



fs'=c- 



o-/r 



f'- h2 
f- h2 



.(ii) 



and so on. 

To calculate M we require the value of <\>' near the surface of the sphere A ; this is due 
to double-sources /i ', m', /* 2 ', /* 3 ', ... at distances c, c-//, c-/ 2 ', c-f 3 ', ... from A, where 
/i '= -27T& 3 , and 



(p-ffi 



and so on. This gives, for points near the surface of A, 



/i' 3 ' 
/s' 3 ' 

J5 J 



.(12) 



Mo 



M2 



4tt0' = (fii + fi 3 ' + fi 6 ' + . . . ) — — - 2 

W (c-/ 2 ) 3 (c-/ 4 ) 3 



+ r- 



J*4 



r cos 6 + &c. 



.(13) 



Hence M= - p J jq>' ^ dS A = P (^ + ^ + fi 6 ' + 



_ a 3 b 3 ( a 3 b 3 a G b 6 ) 

"*" -r \ l+ JFi?=3ft+fi*m-im-tv* J l (14) 



98-99] Motion of Two Spheres 133 

When the ratios ajc and b/c are both small we have 

i=§^a3(l+3^. 3 ), M=2*p^, ^=1^^(1+3^), (15) 

approximately* 

If in the preceding results we put b = a, U' = (J, the plane bisecting AB at right angles 
will be a plane of symmetry, and may therefore be taken as a fixed boundary to the fluid 
on either side. Hence, putting c = 2h, we find, for the kinetic energy of the liquid when a 
sphere is in motion perpendicular to a rigid plane boundary, at a distance h from it, 



result due to Stokes. 



2T=$«pa* (l + §|-3+...) U\ (16) 



99. When the spheres are moving at right angles to the line of centres 
the problem is more difficult ; we shall therefore content ourselves with the 
first steps in the approximation, referring, for a more complete treatment, to 
the papers cited on p. 134. 

Let the spheres be moving with velocities V, V in parallel directions at right angles to 
A, B, and let r, 0, <o and r' f 6', a> be two systems of spherical polar co-ordinates having 
their origins at A and B respectively, and their polar axes in the directions of the velocities 
V, V. The velocity-potential will be of the form 

V<f>+V'<t>', 

with the surface-conditions 

f r = -cos6, ^' = 0, for,=a, (1) 

and |£ = 0, ^=-cos0', for/ = 6 (2) 

If the sphere B were absent the velocity-potential due to unit velocity of A would be 



i a a 
* -s cos 6. 

- r l 



Since r cos 6 = r' cos 6\ the value of this in the neighbourhood of B will be 

a 3 
±- 3 r'cos6', 

approximately. The normal velocity at the surface of B, due to this, will be cancelled by 
the addition of the term 

x a 3 b 3 cos & 

which in the neighbourhood of A becomes equal to 

J-^rcostf, 
nearly. To rectify the normal velocity at the surface of J., we add the term 

x a 6 b 3 cos 6 
*-#- "72~- 

Stopping at this point, and collecting our results, we have, over the surface of A, 

/ o 3 h 3 \ 
<t>=\a>(l +$-jr) cos 6, (3) 

a 3 
and at the surface of B, = |6.— cos#' (4) 

* To this degree of approximation the results may be more easily obtained without the use of 
' images,' the procedure being similar to that of the next Art. 



134 Irrotational Motion of a Liquid [chap, v 

Hence if we denote by P, Q, R the coefficients in the expression for the kinetic energy, 
viz. 

2T=PV* + 2QVV' + R V'\ (5) 

we have *--p//*|£^-|^(l+*^-/ 

«--p//#8>-t^?. < 6 > 

The case of a sphere moving parallel to a fixed plane boundary, at a distance A, is 
obtained by putting b = a, V= V, c=2h, and halving the consequent value of T ; thus 

S2»-iirpo»(l + ftg) F» (7) 

This result, which was also given by Stokes, may be compared with that of Art. 98 (16)*. 

Cylindrical Harmonics. 

100. In terms of the cylindrical co-ordinates x, vr, <o introduced in 
Art. 89, the equation V 2 </> = takes the form 

dx* + dv 2 + *rd*T~ ] ~v*dco 2 ~ V W 

Tnis may be obtained by direct transformation, or more simply by expressing 
that the total flux across the boundary of an element 8x . 8vt . vr 8co is zero, 
after the manner of Art. 83. 

In the case of symmetry about the axis of x, the equation reduces to the 
form (4) of Art. 94. A particular solution is then <j> = e ±kx % (-or), provided 

^(•) + ^W+^W = o ( 2 ) 

This is the differential equation of 'Bessel's Functions' of zero order. Its 
complete primitive consists, of course, of the sum of two definite functions 
of «r, each multiplied by an arbitrary constant. That solution which is finite 
for «■ = is easily found in the form of an ascending series ; it is usually 
denoted by GJ Q (lev;), where 

^o(t)=l-g + 2r^- (3) 

* For a fuller analytical treatment of the problem of the motion of two spheres we refer to 
the following papers: W. M. Hicks, "On the Motion of two Spheres in a Fluid," Phil. Trans. 
1880, p. 455; E. A. Herman, "On the Motion of two Spheres in Fluid," Quart. Journ. Math. 
xxii. (1887) ; Basset, "On the Motion of Two Spheres in a Liquid, &c." Proc. Lond. Math. Soc. 
xviii. 369 (1887). See also C. Neumann, Hydrodynamische Untersuchungen, Leipzig, 1883; 
Basset, Hydrodynamics, Cambridge, 1888. The mutual influence of 'pulsating' spheres, i.e. of 
spheres which periodically change their volume, has been studied by C. A. Bjerknes, with a view 
to a mechanical illustration of electric and other forces. A full account of these researches is 
given by his son Prof. V. Bjerknes in Vorlesungen iiber hydrodynamische Fernkrafte, Leipzig, 
1900-1902. The question is also treated by Hicks, Camb. Proc. iii. 276 (1879), iv. 29 (1880), and 
by Voigt, Gott. Nachr. 1891, p. 37. 



since 



99-100] Cylindrical Harmonics 135 

We have thus obtained solutions of V 2 <£ = of the types* 

£ = e ±te Ji(M (4) 

It is easily seen from Art. 94 (1) that the corresponding value of the stream - 
function is 

^-Tw^/o'N (5) 

The formula (4) may be recognized as a particular case of Art. 89 (6); 
viz. it is equivalent to 

0=1 r e ±k(x+ivrcos^)^ (6) 

Jo(0 = - f r cos(?cos^)^ = - fV C0S *c^, (7) 

7T J TT J 

as may be verified by developing the cosine, and integrating term by term. 

Again, (4) may also be identified as the limiting form assumed by a 
spherical solid zonal harmonic when the order (n) is made infinite, provided 
that at the same time the distance of the origin from the point considered be 
made infinitely great, the two infinities being subject to a certain relation f. 

Thus we may take 

+ -£P.W-(l +£)"*.<«>. (8) 

where we have temporarily changed the meanings of x and <xr, viz. 
r = a + x } vt = 2a sin \ 6 y 

whilst Xn (*r)=i- v 22 y -^ + — - 22 42 M — -^-•••; ( 9 > 

see Art. 85 (4). If we now put k = n/a, and suppose a and n to become 
infinite, whilst k remains finite, the symbols x and -cr will regain their former 
meanings, and we reproduce the formula (4) with the upper sign in the 
exponential. The lower sign is obtained if we start with 

a n+l 

The same procedure leads to an expression of an arbitrary function of m 
in terms of the Bessel's Function of zero order {. According to Art. 88, an 
arbitrary function of latitude on the surface of a sphere can be expanded in 
spherical zonal harmonics, thus 

F(n) = t(n + i)P n (ri ^FW)P n {p')dp' (10) 

* Except as to notation these solutions are to be found in Poisson, I.e. ante p. 18. 
+ This process was indicated, without the restriction to symmetry, by Thomson and Tait, 
Art. 783 (1867). 

% The procedure appears to be due substantially to C. Neumann (1862). 



136 Irrotational Motion of a Liquid [chap, v 

If we denote by ct the length of the chord drawn to the variable point from 
the pole (0 = 0) of the sphere, we have 

tjs = 2a sin J 0, ^8sr = — a 2 Bfju, 

where a is the radius, so that the formula may be written 

f{m)-\*(n + \)H % {w)\*f{J)P n {*)v/dm/ (11) 

a Jo 

n 1 

If we now put k = - , 8k — - , 

r a a 

and finally make a infinite, we are led to the important theorem*: 

/(*r)= I™ J (kv)kdk \™ f(v l )J*(W)'a'dv' (12) 

Jo Jo 

101. If in (1) we suppose </> to be expanded in a series of terms varying 
as cos sco or sinsa), each such term will be subject to an equation of the form 

S+S+ i S*-A+-o < 13 > 

dor 9ot 2 ta d^r n 2 
This will be satisfied by <j> = e ±kx % (ct), provided 

rt"(»)+iX'(-) + (*-5)x(»)-o> a*) 

which is the differential equation of Bessel's Functions of order sf. The 
solution which is finite for w — may be written % (ot) = CJ S (kxs), where 

Js (?) = 2 s . n (J) J 1 " 2 (2* + 2) + 2 . 4 (2 5 + 2) (2* + 4) " ' * j* '" (15) 

The complete solution of (14) involves, in addition, a Bessel's Function 
'of the second kind' with whose form we shall be concerned at a later period 
in our subject f. 

We have thus obtained solutions of the equation V 2 <£ = 0, of the types 
$ = e± k *J s (k>*) C HsG> (16) 

* For more rigorous proofs, and for the history of the theorem, see Watson, I.e. infra. 

t Forsyth, Art. 100; Whittaker and Watson, c. xvii. 

X For the further theory of the Bessel's Functions of both kinds recourse may be had to 
Gray and Mathews, Treatise on Bessel Functions, 2nd ed., London, 1922, and to G. N. Watson, 
Theory of Bessel Functions, Cambridge, 1923, where ample references are given to previous writers. 
An account of the subject, from the physical point of view, will be found in Eayleigh's Theory 
of Sound, cc. ix., xviii., with many important applications. 

Numerical tables of the functions J s (f ) have been constructed by Bessel and Hansen, and more 
recently by Meissel (Berl. Abh. 1888). These are reproduced by Gray and Mathews, and, with 
valuable extensions, in Watson's treatise. Abridged tables are included in the collections of Dale 
and of Jahnke and Emde referred to on p. 114. 



100-102] Cylindrical Harmonics 137 

These may also be obtained as limiting forms of the spherical solid harmonics 
r n t-» , x cos) a n+1 „ . , x cos) 

a n \rs gin j . rn+ i n \t»j gm j 

with the help of the expansion (6) of Art. 86*. 

102. The formula (12) of Art. 100 enables us to write down expressions, 
which are sometimes convenient, for the value of <£ on one side of an infinite 
plane (x — 0) in terms of the values of <j> or dcp/dn at points of this plane, in 
the case of symmetry about an axis (Ox) normal to the planef. Thus if 

<t> = F(er), for# = 0, (1) 

we have, on the side x > 0, 

a >= I™ e - kx Jo(kvr)kdk \" F(w')J (hm')w'dm i (2) 

Jo Jo 

Again, if -^^/(^ for# = 0, (3) 

we have <£ = f°V te J (k*r)dk f °" f(^')J (k^ , )^'d'm' (4) 

Jo Jo 

The exponentials have been chosen so as to vanish for x = oo . 

Another solution of these problems has already been given in Art. 58, 

from equations (12) and (11) of which we derive 



#sS)« (5) 



and 



/: 



*--l\\tf <«> 

respectively, where r denotes distance from the element BS of the plane to 

the point at which the value of <f> is required. 

We proceed to a few applications of the general formulae (2) and (4). 

1°. If, in (4), we assume /(or) to vanish for all but infinitesimal values of c;, and to 
become infinite for these in such a way that 

/(or) 27TZ(7C?G7 = ^. 

we obtain 4tt$= / e- kx J Q {hw)dk i (7) 

J o 

and therefore, since Jo'= —J\, 

4:7r^t=-w\ e~ kx J^km) dk, (8) 

by Art. 100 (5). 

* The connection between spherical surface-harmonics and Bessel's Functions was noticed by 
Mehler, " Ueber die Vertheilung d. statischen Elektricitat in einem v. zwei Kugelkalotten begrenzten 
Korper," Crelle, lxviii. (1868). It was investigated independently by Eayleigh, "On the Eelation 
between the Functions of Laplace and Bessel," Proc. Lond. Math. Soc. ix. 61 (1878) [Papers, i. 338] ; 
see also Theory of Sound, Arts. 336, 338. 

There are also methods of deducing Bessel's Functions 'of the second kind' as limiting 

forms of the spherical harmonics Q n (a), Q n 8 (^) C ° S j- sw; for these see Heine, i. 184, 232. 
f The method may be extended so as to be free from this restriction. 



138 Irrotational Motion of a Liquid [chap, v 

By comparison with the primitive expressions for a point-source at the origin (Art. 95), 
we infer that 

f" e~ 1cX J (km)dk=-, f e~ kx J 1 (km)dk= ,™ . , (9) 

where r=J(x 2 + m 2 ) ; these are in fact known results* 

2°. Let us next suppose that sources are distributed with uniform density over the 
plane area contained by the circle m=a, #=0. Using the series for J , J ly or otherwise, 
we find 

( a j Q {kw)wdvy= < ^J 1 {ka) (10) 

Hence t 

<£ = — re-xxJoi^J^ka) — , yj,= -— re-teJ^k^J^ka)^, (11) 

ttolj o lc ira J o lc 

where the constant factor has been chosen so as to make the total flux through the circle 
equal to unity. 

3°. Again, if the density of the sources, within the same circle, vary as ll>J(a 2 -m 2 ), 
we have to deal with the integral J 

|V (to)- 7 ^^ ) = a | o i V (tosin3)sin5^ = ^, (12) 

where the evaluation is effected by substituting the series form of J , and treating each 
term separately. Hence 



JL 1 f *» r n s ■ 7 dk , m r fa T „ , . , dk 

9 = ~ — / e- kx J (km)smka^ r , \i,= -- — / «-« J x (km) sin fca-j-, 

Ana J o K ZrraJ q K 



.(13) 



if the constant factor be determined by the same condition as before §. 

It is a known theorem of Electrostatics that the assumed law of density makes </> 
constant over the circular area. It may be shewn independently that 



J (km) smka- r =%ir i or sin -1 — , ) 

h W \ (14) 

T n . . , dk a-s/(a 2 -z3 2 ) a 

Ji (km) sin ka-j-= — , or — 

o k m m 



according as m<a\\. The formulae (13) therefore express the flow of a liquid through a 
circular aperture in a thin plane rigid wall. Another solution will be obtained in Art. 108. 
The corresponding problem in two dimensions was solved in Art. 66, 1°. 

4°. Let us next suppose that when x=0, we have <j> = C *J(a 2 -m 2 ) for m <a, and = 
for m > a. We find 

J Jo(km)s/(a 2 -m^)mdm=a s t " J (ka sin S) sin $ cos 2 Sd$ = a 3 ^i (ka), ...(15) 
provided ^ l(C )^( 1 _^ + ^ 577 _...) = .^^ (16 ) 

Hence, by (2), <£= - cj%-» J (km) J* ( S -^ 

* The former is due to Lipschitz, Crelle, lvi. 189 (1859); see Watson, p. 384. The latter 
follows by differentiation with respect to m and integration with respect to x. 

t Cf . H. Weber, Crelle, lxxv. 88 ; Heine, ii. 180. 

X The formula (12) has been given by various writers; see Eayleigh, Papers, iii. 98; Hobson, 
Proc. Lond. Math. Soc. xxv. 71 (1893). 

§ Cf. H. Weber, Crelle, lxxv. (1873) ; Heine, ii. 192. 

|| H. Weber, Crelle, lxxv.; Watson, p. 405. See also Proc. Lond. Math. Soc. xxxiv. 282. 



dk (17) 



102-103] Ellipsoidal Harmonics 139 

This gives, for # = 0, 

-(jP) =C [" J (km) am ka^+Cw j"° J ' (km) sin kadk, (18) 

after a partial integration. The value of the former integral is given in (14), and that of 
the latter can be deduced from it by differentiation with respect to m. Hence 



-S^^-K^-^-^M' (19) 



according as w < a. It follows that if 0=2/*- . U, the formula (17) will relate to the motion 
of a thin circular disk with velocity U normal to its plane, in an infinite mass of liquid. 
The expression for the kinetic energy is 

2T= - P fU^dS=7rpC 2 r^a 2 -™ 2 ) 2irmdm=$7r 2 pa*C\ 

or 2T=§pa,3U 2 (20) 

The effective addition to the inertia of the disk is therefore 2/?r ( = '6366) times the 
mass of a spherical portion of the fluid, of the same radius. For another investigation of 
this question, see Art. 108. 

Ellipsoidal Harmonics. 

103. The method of Spherical Harmonics can also be adapted to the 
solution of the equation 

W=o, (i) 

under boundary-conditions having relation to ellipsoids of revolution*. 
Beginning with the case where the ellipsoids are prolate, we write 
x — h cos 6 cosh 77 = kfi £, y — ^ cos co, z = -or sin co, ] 
where sr = A?sin d sinh 77 = k(\ -,*»)*(£»- 1)*. J'" < 2 ' 

The surfaces f= const., /n = const, are confocal ellipsoids and hyperboloids 
of two sheets, respectively, the common foci being the points (+ k, 0, 0). The 
value of f may range from 1 to 00 , whilst fx lies between ± 1. The co-ordinates 
ft, £ co form an orthogonal system, and the values of the linear elements Bs^, 
8s f, Sso, described by the point (%, y, z) when fi, f, co separately vary are 

(3) 

To express (1) in terms of our new variables we equate to zero the total 
flux across the walls of a volume element Bs^Bs^Bs^, and obtain 

or, on substitution from (3), 

* Heine, "Ueber einige Aufgaben, welche auf partielle Differentialgleichungen fuhren," 
Crelle, xxvi. 185 (1843), and Kugelfunctionen, ii. Art. 38. See also Ferrers, c. vi. 



140 Irrotational Motion of a Liquid [chap, v 

This may also be written 

d \a ^ 8 *1 i '*■ y +-» fa m d +} ■ x 9 ^ (4) 

3 M 1 (1 * } a M | + i - ff do,* ~ i% r ? } art + rrp g^-2 w 

104. If <f> be a finite function of /z. and co, from /*=— lto/Lt = + l and 
from co = to <w = 27r, it may be expanded in a series of surface harmonics of 
integral orders, of the types given by Art. 86 (7), where the coefficients are 
functions of f ; and it appears on substitution in (4) that each term of the 
expansion must satisfy the equation separately. Taking first the case of the 
zonal harmonic, we write 

+ -p n (r).z, (5) 

and on substitution we find, in virtue of Art. 84 (1), 

M {1 -^ d i\ +n(n+1)Z ^ ' (6) 

which is of the same form as the equation referred to. We thus obtain the 
solutions 

*-P,0«).P„(t), (7) 

and +-P»6»)-.ft.(tX ( 8 ) 

where 

C»(f)=Pn(t) ] f {p n(f )Jl(fl_l)' 

= ^ ! Wi . (n + l)(n + 2) n _ 3 

1.3...(2n + l) ( fe 2(2^ + 3) * 

(»+l)(n + 2)(n + 3)(n+4) 5 1 , gv * 

* 2.4(2n + 3)(2n + 5) * J* " ; 

The solution (7) is finite when f = 1, and is therefore adapted to the space 
within an ellipsoid of revolution; while (8) is infinite for £= 1, but vanishes 
for f = oo , and is therefore appropriate to the external region. As particular 
cases of the formula (9) we note 

« a (D = i(3? 2 -i)iog|^-J-ir. 

The definite-integral form of Q n shews that 

^.(o^p-^^e.«)— jfii (io) 

The expressions for the stream-function corresponding to (7) and (8) are 
readily found; thus, from the definition of Art. 94, 

* Ferrers, c. v.; Todhunter, c. vi.; Forsyth, Arts. 96-99. 



103-106] Motion of an Ovary Ellipsoid 141 

g = -^-l)|, Sf-*(1-*>|* d2) 

Thus, in the case of (7), we have 

~~n(n + l) { * _1) df "d/i \ {i fl) dp 

wh — *-^ ) ( 1 -^^^ Ll >^r (13) 

The same result will follow of course from the second of equations (12). 
In the same way, the stream-function corresponding to (8) is 

+-^< 1 -^^-<« , - 1 )^ Q < 14) 

105. We can apply this to the case of an ovary ellipsoid moving parallel 
to its axis in an infinite mass of liquid. The elliptic co-ordinates must be 
chosen so that the ellipsoid in question is a member of the confocal family, 
say that for which f = f . Comparing with Art. 103 (2) we see that if a, c be 
the polar and equatorial radii, and e the eccentricity of the meridian section, 
we must have 

k = ae, & = !/«, k(tf-l)$ = c. 

The surface-condition is given by Art. 97 (1), viz. we must have 

^ = - J EW(1 -/*■)({*-!) + const., (^ 

for f= f - Hence putting n = l in Art. 104 (14), and introducing an arbitrary 
multiplier A, we have 

K 



with the condition 



^=^(l-^)(?*-l){|log|±| 



The corresponding formula for the velocity-potential is 



■(2) 
.(3) 



#-^JKlogj£|--l} (4) 

The kinetic energy, and thence the inertia-coefficient due to the fluid, 
may be readily calculated by the formula (5) of Art. 94. 

106. Leaving the case of symmetry, the solutions of V 2 <f> = when <j> is a 
tesseral or sectorial harmonic in fju and <w are found by a similar method to be 
of the types 

0=p„«( /t ).p M »(?)^J sa)) a) 

<t> = Pn°(ri.Qn>(!;)'**\s a >, (2) 



142 Irrotational Motion of a Liquid [chap, v 

where, as in Art. 86, P n ° (fi) = (1 - fff* - "ftffi , (3) 

whilst (to avoid imaginaries) we write 

JVC?)^ 2 -!)* 8 *!^, (4) 

It may be shewn that 

enS(r)=(_)8 (^^yi PnS(r) 'J f (p,/(?)p.(r 2 -i)' (6) 

*_ p, ( „*p-^» (! , ( „. ( -)«|±|: ? L I m 

As examples we may take the case of an ovary ellipsoid moving parallel 
to an equatorial axis, say that of y, or rotating about this axis. 

1°. In the former case, the surface-condition is 

d±__ v dy 

for £= £o> where Fis the velocity of translation, or 

j* — F. * & . .(1-^coso, (8) 

This is satisfied by putting n = 1, * = 1, in (2), viz. 

^, = 4(l- M a )i(? 2 -l)*.jjlog|±j-^ I )cos6, ; (9) 

the constant A being given by 

4 lo 4S-J^}=-^ (io > 

2°. In the case of rotation about Oy, if £l y be the angular velocity, we 
must have 

30 _ n / dx dz\ 

for?=r o or ^=k*n y . — 1 /.(l-^sino) (11) 

Putting n = 2, s = 1, in the formula (2) we find 

f=4 /i (l-^)i(^-l)ijfriog|±|-3-^Jsin &)> ...(12) 

J. being determined by comparison with (11). 

107. When the ellipsoid is of the oblate or ' planetary ' form, the appropriate 
co-ordinates are given by 

x = k cos 6 sinh 77 = &//,£ ?/ = «rcosG>, ^ = -or sin co, } ,_. 

where «• - A; sin (9 cosh 77 = k (1 - a* 2 )*(? 2 + 1)*. J 



106-107] Motion of a Planetary Ellipsoid 143 

Here f may range from to oo (or, in some applications, from — oo through 
to 4- oo ), whilst /ub lies between ± 1. The quadrics f = const., /a = const, are 
planetary ellipsoids and hyperboloids of revolution of one sheet, all having 
the common focal circle x = 0, ct = k. As limiting forms we have the ellipsoid 
f = 0, which coincides with the portion of the plane x = for which «r < k, and 
the hyperboloid /z = coinciding with the remaining portion of this plane. 
With the same notation as before we find 

(2) 

and the equation of continuity becomes 

a^O 1 ^a^j arr 3?J (i-/* 2 )(r 2 + i)ao> 2 u ' 



or 



lr{a-^» + r^S-il(«-+l)» + ^S..^) 



3yL6 ( 3/X 



l-/x 2 3a> 2 3f ( V5 7 afJ ' £ 2 + l3a> 2 

This is of the same form as Art. 103 (4), with i% in place of f, and the like 
correspondence will run through the subsequent formulae. 

In the case of symmetry about the axis we have the solutions 

*-P.0»).JPto(ft W 

and *.«i > «G»).9»(?). (5) 

n(n-l)(>i-2)(n-3) ) 

+ 2.4(2n-l)(2n-3) & )' "' W 

and ^(?)=F»(0j d ^ (0|2(?2+jr 

= " ! I^-n-i _ (* + l)(n + 2) ._ n _ 3 

1.3.5.-.(2JH-l)-l i 2(2^ + 3) & 

, (n + l)(n+2)(n + 3)(w + 4) 



2.4(2n + 3)(2n + 5) ? " ' '" \ ' (7) 

the latter expansion being however convergent only when f > 1 *. As before, 
the solution (4) is appropriate to the region included within an ellipsoid of 
the family f = const., and (5) to the external space. 

We note that p n (0 ^ - ^1> ?>l (r) = _ _i_ ( 8 ) 

As particular cases of the formula (7) we have 

g (r)=cot-i?, gi (?) = l-rcot- 1 r > 

? 2 (r)=*(3r 2 +])cot- i c-K. 

* The reader may easily adapt the demonstrations referred to in Art. 104 to the present case. 



144 Irrotational Motion of a Liquid [chap, v 

The formulae for the stream-function corresponding to (4) and (5) are 

and +-^< 1 -*>^-« i+1 >^ do 

108. 1°. The simplest case of Art. 107 (5) is when n= 0, viz. 

</> = ^lcot- 1 ^ (1) 

where f is supposed to range from — oo to + oo . The formula (10) of the 
last Art. then assumes an indeterminate form, but we find by the method of 
Art. 104, 

ir = Akp, (2) 

where /m ranges from to 1. This solution represents the flow of a liquid 
through a circular aperture in an infinite plane wall, viz. the aperture is the 
portion of the plane yz for which m < k. The velocity at any point of the 
aperture (f =0) is 

_ _ 1 <ty = A 

ta dco ~ (p _ w 2)£ ' 

since, when oo — 0, kfi — (k 2 — -sr 2 )^. The velocity is therefore infinite at the 
edge. Compare Art. 102, 3°. 

2°. Again, the motion due to a planetary ellipsoid (?=f ) moving with 
velocity U parallel to its axis in an infinite mass of liquid is given by 

^^(Wcot- 1 ^ ^=i^(l-^ 2 )(? 2 + l){^ T -cot- 1 rf,...(3) 

where A = - k If + Ww ~ cot_1 ?o 

Denoting the polar and equatorial radii by a and c, and the eccentricity of 
the meridian section by e, we have 

In terms of these quantities 

A = - Uc ^\(l-e 2 )i-~ sin- 1 e\ (4) 

The forms of the lines of motion, for equidistant values of yjr, are shewn 
on the next page. Cf. Art. 71, 3°. 

The most interesting case is that of the circular disk, for which e = 1, 
and A = 2Uc/7r. The value of <f> given in (3) becomes equal to ±A/jl, or 
+ A (1 — -g^/c 2 )^, for the two sides of the disk, and the normal velocity to 
± U. Hence the formula (4) of Art. 44 gives 

2T=$pc*U*, (5) 

as in Art. 102 (20). 



107-109] 



Motion of a Planetary Ellipsoid 



145 




X' 



X 




109. The solutions of the equation Art. 107 (3) in tesseral harmonics are 
*-Pn'(ri.tf(fi.*$«» 9 (1) 

and *-^n < 0*).9n , (f)-S«», (2) 

Where Pn 8 (f) = (r 2 +l)* 8 ^f|P, (3) 

and *.'«?)-({* + 1)**^P, 

K ' (n-s)rPn W-) t ift .(f)J«(fi+i) <*> 

These functions possess the property 

*•"' <*r if ?«(?)-( ) (M _ g)!?a+1 (5) 

We may apply these results as in Art. 108. 



146 Irrotational Motion of a Liquid [chap, v 

1°. For the motion of a planetary ellipsoid (f = f ) parallel to the axis of 
y we have n = 1, s = 1, and thence 



.(6) 



<j> = A (1 - ft (f 2 + 1)* {^ - cot- 1 fj cos o>, 

with the condition — = — V ~ , 

for f = f , F" denoting the velocity of the solid. This gives 

A \$rh)-***<\-- hY [ (7) 

In the case of the disk (f = 0), we have A = 0, as we should expect. 

2°. Again, for a planetary ellipsoid rotating about the axis of y with 
angular velocity f\ , we have, putting n — 2, 5 = 1, 

^=^(l- / a 2 )i(r 2 +l)i|3rcot- 1 r-3 + ^i- T } S ina )) (8) 

with the surface-condition 

d<j> _ n / 3d? cte 

at — 1J *raf~*af. 

= _^rii L is . nw 

For the circular disk (£b = 0) this gives 

firA^-Mly (10) 

At the two surfaces of the disk we have 

<j> = + 2A/JL (1 - yu 2 )* to *>, ^ = + kCly (1 - iff sin », 
and substituting in the formula 



2T=-p [U^vdvdu, 
we obtain IT = Jf pc 5 . O y 2 (11)* 

110. In questions relating to ellipsoids with three unequal axes we may 
employ the more general type of Ellipsoidal Harmonics, usually known by the 
name of 'Lame's Functions f.' Without attempting a formal account of these 
functions, we will investigate some solutions of the equation 

V 2 <£ = 0, (1) 

in ellipsoidal co-ordinates, which are analogous to spherical harmonics of the 
first and second orders, with a view to their hydrodynamical applications. 

* For further solutions in terms of the present co-ordinates see Nicholson, Phil. Trans. A, 
ccxxiv. 49 (1924). 

t See, for example, Ferrers, Spherical Harmonics, c. vi.; W. D. Niven, Phil. Trans. A, 
clxxxii. 182 (1891) and Proc. Roy. Soc. A, lxxix. 458 (1906); Poincar^, Figures d'Equilibre d'une 
Masse Fluide, Paris, 1902, c. vi.; Darwin, Phil. Trans. A, cxcvii. 461 (1901) [Scientific Papers, 
Cambridge, 1907-11, iii. 186]; Whittaker and Watson, c. xxiii. An outline of the theory is given 
by Wangerin, I.e. ante p. 110. 



109-1 10J Motion of a Planetary Ellipsoid 147 

It is convenient to prefix an investigation of the motion of a liquid con- 
tained in an ellipsoidal envelope, which can be treated at once by Cartesian 
methods. 

Thus, when the envelope is in motion parallel to the axis of x with 
velocity U, the enclosed fluid moves as a solid, and the velocity-potential is 
simply <£ = — Ux. 

Next let us suppose that the envelope is rotating about a principal axis 
(say that of x) with angular velocity £l x . The equation of the surface being 



- + ^+-=1 (2) 

a 2+ 6 2+ c 2 A, W 



the surface-condition is 



~a 2 dx~ b 2 dy c 2 hz~ & ll * z + ^W 

We therefore assume (j> — Ayz, which is evidently a solution of (1), and obtain, 
on determining the constant by the condition just written, 

b 2 -c 2 

Hence, if the centre be moving with a velocity whose components are 
U, V, W and if H Xi Cl y , £l z be the angular velocities about the principal axes, 
we have by superposition* 

b 2 — c 2 c 2 - a 2 a 2 — b 2 

$ = - Ux - Vy - Wz --^—- 2 n x yz - -^— 2 n y zx - - z —^n z xy. ...(3) 

We may also include the case where the envelope is changing its form, 
but so as to remain ellipsoidal. If in (2) the lengths (only) of the axes are 
changing at the rates a, b, c, respectively, the general boundary-condition, 
Art. 9 (3), becomes 

a^+b^ + c^+atdx + Vdy + ctdz- ' W 

which is satisfied f by 

♦-■-tg'+jii'+j*) W 



The equation (1) requires that 



& - + b 7 + ° = (6) 

a b c v 7 



which is in fact the condition which must be satisfied by the changing ellip- 
soidal surface in order that the enclosed volume (%irabc) may be constant. 

* This result appears to have been published independently by Beltrami, Bjerknes, and 
Maxwell, in 1873. See Hicks, "Beport on Kecent Progress in Hydrodynamics," Brit. Ass. Rep. 
1882, and Kelvin's Papers, iv. 197 (footnote). 

t C. A. Bjerknes, " Verallgemeinerung des Problems von den Bewegungen, welche in einer 
ruhenden unelastischen Fliissigkeit die Bewegung eines Ellipsoids hervorbringt," Gottinger 
Nachrichten, 1873, pp. 448, 829. 



148 Irrotational Motion of a Liquid [chap, v 

111. The solutions of the corresponding problems for an infinite mass of 
fluid bounded internally by an^ellipsoid involve the use of a special system of 
orthogonal curvilinear co-ordinates. 

If x, y y z be functions of three parameters X, fi, v, such that the surfaces 

X = const., fjL — const., v — const (1) 

are mutually orthogonal at their intersections, and if we write 



h x 2 ~ \d\J ^{dxj + \dx) ' 

_1 (dx\ 2 (dy\ 2 (dz\ 2 
% \o"/ vW \<W 



•(2) 



the direction-cosines of the normals to the three surfaces which pass through 
(x, y, z) will be 

(^!'*!'*»i9- (^I'^l'* 8 !)' ( A '£-^l' A3 a-3'- (3) 

respectively. It follows that the lengths of linear elements drawn in the 
directions of these normals will be 

BX/h 1} BfM/h 2} Bv/h 3 . 

Hence if $ be the velocity-potential of a fluid motion, the total flux 
into the rectangular space included between the six surfaces X ± ^8X, //, ± -JS/*, 
v ± 1 8v will be 

It appears from Art. 42 (3) that the same flux is expressed by V 2 <£ multiplied 
by the volume of the space, i.e. by 8x8/jL8v/h 1 h 2 h 3 . Hence* 

^-^{k(&S?) + 5(oc© + s(&S}'- - (4) 

Equating this to zero, we obtain the general equation of continuity in 
orthogonal co-ordinates, of which particular cases have already been investi- 
gated in Arts. 83, 103, 107. 

The theory of triple orthogonal systems of surfaces is very attractive 
mathematically, and abounds in interesting and elegant formulae. We may 
note that if X, fi, v be regarded as functions of x, y, z, the direction-cosines 

* The above method was given in a paper by W. Thomson, " On the Equations of Motion of 
Heat referred to Curvilinear Co-ordinates," Camb. Math. Journ. iv. 179 (1843) [Papers, i. 25]. 
Reference may also be made to Jacobi, "Ueber eine particulate Losung der partiellen Diffe- 
rentialgleichung ," Crelle, xxxvi. 113 (1847) [Werke, ii. 198]. 

The transformation of V 2 to general orthogonal co-ordinates was first effected by Lame, " Sur 
les lois de l'equilibre du fluide £th£re," Journ. de VEcole Polyt. xiv. 191 (1834). See also his Legons 
sur les Coordonne.es Curvilignes, Paris, 1859, p. 22. 



111-112] 



Orthogonal Co-ordinates 



149 



of the three line-elements above considered can also be expressed in the 
forms 

1 3X 1 3X 1 3X\ /l dfi I dp 1 d/i\ /l dv 1 dv 1 9i/> 



9-0 



V^i 3# ' h-idy' Ai 3.sv ' \A 2 3# ' h 2 dy ' h 2 dz J ' \h 3 dx ' h z dy' h z dz/ 

(5) 

from which, and from (3), various interesting relations can be inferred. The 
formulae already given are, however, sufficient for our present purpose. 

112. In the applications to which we now proceed the triple orthogonal 
system consists of the confocal quadrics 



x 2 y 2 z 2 

W+e + v + e + ?+0 



-1 = 0, 



.(i) 



whose properties are explained in books on Solid Geometry. Through any 
given point (x, y, z) there pass three surfaces of the system, corresponding 
to the three roots of (1), considered as a cubic in 6. If (as we shall for the 
most part suppose) a > b > c, one of these roots (X, say) will lie between oo 
and - c 2 , another (fi) between — c 2 and — 6 2 , and the third (v) between — b 2 
and — a 2 . The surfaces X, //,, v are therefore ellipsoids, hyperboloids of one 
sheet, and hyperboloids of two sheets, respectively. 

It follows immediately from this definition of X, fi, v, that 



+ 



ft 



z 2 (\-O)Qi-6)(i>-0) 

c 2 + 6 (a 2 + 0)(b 2 + d)(c 2 + 0) 



•(2) 



a 2 +d b 2 + 

identically, for all values of 6. Hence multiplying by a 2 + 6, and afterwards 
putting 6 = — a 2 , we obtain the first of the following equations : 

(a 2 +X)(a 2 + ^)(a 2 + ^ 

(a 2 -6 2 )(a 2 -c 2 ) 

(b 2 + \)(b 2 +ri(b 2 + v) 

(b 2 -<?)(b 2 -a 2 ) 



2/* = 



These give 



dx _ x x 
3X~~ 2 (?Tx' 3X 



(c 2 + \)(c 2 + fjL)(c 2 + v) 
(c 2 -a 2 )(c 2 -b 2 ) 

ty i y dz 



i 



6 2 + X' 3X 
and thence, in the notation of Art. ill (2), 



-* 



c 2 + X' 



.(4) 



1 , ( a* y 2 

h x 2 ±}( a 2 + X) 2 + (6 2 + X) 2 



+ 



.(5) 



(c 2 + X) 2 

If we differentiate (2) with respect to 6 and afterwards put = X, we deduce 
the first of the following three relations : 

(a 2 + X)(6 2 + X)(c 2 + X) * 



hx 2 — 4 

(X — /jl) (X — v) 

Aa 2_ 1 , (^ + ^(^ + ^)(c 2 + ^) 

{a 2 + v)(b 2 + v){c 2 + v) 



A 3 2 = 4 



(i/ - X) (v - /a) 



.(6) 



150 Irrotational Motion of a Liquid [chap, v 

The remaining relations of the sets (3) and (6) have been written down from 
symmetry*. 

Substituting in Art. Ill (4), we find -J- 

v '» — ( M - y )(,-x)(x-^) [»-'){^+^( y + ^^ + ^^)' 

+ (» - X) |(a 2 + /*)* (6 2 + p)i (c 2 + M )* |- 

+ (X - /i) |(a 2 + *)* (ft 2 + k)* (c 2 + 1,)* I!" 

..:... (7) 

113. The particular solutions of the transformed equation V 2 c/> = which 
first present themselves are those in which (f> is a function of one (only) of 
the variables X, fx, v. Thus <f> may be a function of X alone, provided 

(a 2 + X)* (b 2 + X)* (c 2 + \)i g£ = const., 

whence < / > = C f -r- , (1) 

if A = {(a 2 + X)(6 2 + \)(c 2 + X)}*, (2) 

the additive constant which attaches to <j> being chosen so as to make (f> 
vanish for X — oo . 

In this solution, which corresponds to <\> = A/r in spherical harmonics, 
the equipotential surfaces are the confocal ellipsoids, and the motion in the 
space external to any one of these (say that for which \ = 0) is that due to a 
certain arrangement of simple sources over it. The velocity at any point is 
given by the formula 

-* 2 f=4 ••••• < 3 > 

At a great distance from the origin the ellipsoids \ become spheres of 
radius \^, and the velocity is therefore ultimately equal to 20/r 2 , where r 
denotes the distance from the origin. Over any particular equipotential 
surface X, the velocity varies as the perpendicular from the centre on the 
tangent plane. 

To find the distribution of sources over the surface X = which would 
produce the actual motion in the external space, we substitute for <f> the 
value (1), in the formula (11) of Art. 58, and for <j>' (which refers to the 
internal space) the constant value 

*'=<! w 

* It will be noticed that h lt h 2 , h 3 are double the perpendiculars from the origin on the 
tangent planes to the three quadrics X, ji, v. 

+ Cf. Lam6, "Sur les surfaces isothermes dans les corps solides homogenes en ^quilibre de 
emperature," Liouville, ii. 147 (1837). 



H2-H3] Ellipsoidal Co-ordinates 151 

The formula referred to then gives, for the surface-density of the required 
distribution, 

JL h > • -< 5) 

The solution (1) may also be interpreted as representing the motion due 
to a change in the dimensions of the ellipsoid, such that the surface remains 
similar to itself, and retains the directions of its principal axes unchanged. 
If we put 

a/a = 6/6 = c/c, = k, say, 

the surface-condition Art. 110 (4) becomes 

— d<j>/dn = \kh 1 , 
which is identical with (3), if we put \ — 0, G = ^kabc. 

A particular case of (5) is where the sources are distributed over the 
elliptic disk for which A, = — c 2 , and therefore z 2 = 0. This is important in 
Electrostatics, but a more interesting application from the present paint of 
view is to the flow through an elliptic aperture, viz. if the plane xy be 
occupied by a thin rigid partition with the exception of the part included by 
the ellipse 

x XI 

-* + ^ = l> —0, 

we assume, putting c = in previous formulae, 

* = :M JoV + X)4J + X)4x* (6) 

where the upper limit is the positive root of 

and the negative or the positive sign is to be taken according as the point for 
which <j> is required lies on the positive or the negative side of the plane xy. 
The two values of <j> are continuous at the aperture, where \ = 0. As before, 
the velocity at a great distance is equal to 2A/r 2 , and the total flux through the 
area 2-rrr 2 is therefore 4tirA. The total range of <j> from X— — oo to X = + oo is 

2A r * -44 \ ¥ d0 

J o (a 2 + X)* (6 2 + X)* X* J o \/(a 2 sin 2 6 + 6 2 cos 2 6) 

The 'conductivity/ therefore, of the aperture (to borrow a term from elec- 
tricity) is 

^•Jo V(a 2 sin 2 + 6 2 cos 2 0) W 

For a circular aperture this = 2a. 

For points in the aperture the velocity may be found immediately from 
(6) and (7); thus we may put 

x ^lA x 2 y 2 \* ^ _2A\* 



152 Irrotational Motion of a Liquid [chap, v 

approximately, since X is small, whence 

-S-S'-('-S-$f <»> 

This becomes infinite, as we should expect, at the edge. The particular case 
of a circular aperture has already been solved otherwise in Arts. 102, 108. 

114. We proceed to investigate the solution of V 2 ^> = 0, finite at infinity, 
which corresponds, for the space external to the ellipsoid, to the solution 
cf) = x for the internal space. Following the analogy of spherical harmonics 
we may assume for trial 

* = «X» (!) 

which gives V 2 ^ + -^ = 0, (2) 

X OX 

and inquire whether this can be satisfied by making % equal to some function 
of X only. On this supposition we shall have, by Art. Ill, 

d X_ h d X h fa 

te~ Ux' ni dx' 

and therefore, by Art. 112 (4), (6), 

2d X = i (b* + X)(c* + X) d X ^ 
xdx (X — fi) (X —v) dX' 

On substituting the value of V 2 ^ in terms of X, the equation (2) becomes 

S 2 X = -(^)(c* + X)g 



f(a» + A.)* (6* + X)* (e» + X)* iV x = - (6 2 + X) (c* + X) & , 



which may be written 



- log {(a 2 + X)* (6 2 + X)* (c 2 + X)* ^d = - 



dX & ( ' . . : ^Xj a 2 + X* 

Hence % = cf°° s ^— i i> (3) 

* Ja (a 2 + X)*(6 2 +X)*(c 2 + X)* w 

the arbitrary constant which presents itself in the second integration being 
chosen as before so as to make x vanish at infinity. 

The solution contained in (1) and (3) enables us to find the motion of a 
liquid, at rest at infinity, produced by the translation of a solid ellipsoid 
through it, parallel to a principal axis. The notation being as before, and the 
ellipsoid 

a 2+ b 2 + c 2 W 

being supposed in motion parallel to x with velocity U, the surface- 
condition is 

K~*S- fo ^=° < 5 > 



H3-H4] Translation of an Ellipsoid 153 

Let us write, for shortness, 

r°° d\ r°° d\ r°° d\ 

a ° = a H(^Tx)A' A = a6c J (^Tx)A' ^ = a6c J (?TxyA' 

(6) 

where A = {(a 2 + X) (6 2 + X) (c 2 + X)}* (7) 

It will be noticed that these quantities a , /3 , 7o are purely numerical. The 
conditions of our problem are satisfied by 

*-*JT(??W (8) 

provided Q = jbc_TJ. (9) 

2 - a 

The corresponding solution when the ellipsoid moves parallel to y or z can 
be written down from symmetry, and by superposition we derive the case 
where the ellipsoid has any motion of translation whatever*. 

At a great distance from the origin, the formula (8) becomes equivalent to 

+-*c£. (10) 

which is the velocity-potential of a double source at the origin, of strength 

|7r(7, or 

compare Art. 92. 

The kinetic energy of the fluid is given by 

where I is the cosine of the angle which the normal to the surface makes 
with the axis of x. Since the latter integral is equal to the volume of the 
ellipsoid, we have 

ZT-^.frabcp.U* (11) 

The inertia-coefficient is therefore equal to the fraction 

*-s2j; (12) 

of the mass displaced by the solid. For the case of the sphere (a = b = c) we 
find ao = f > & = J> in agreement with Art. 92. If we put a = 6, we get the 
case of an ellipsoid of revolution. 

* This problem was first solved by Green, "Eesearches on the Vibration of Pendulums in 
Fluid Media," Trans. R. S. Edin. xiii. 54 (1883) [Papers, p. 315]. The investigation is much 
shortened if we assume at once from the Theory of Attractions that (8) is a solution of V 2 = O, 
being in fact (except for a constant factor) the ^-component of the attraction of a homogeneous 
ellipsoid at an external point. 



154 Irrotational Motion of a Liquid [chap, v 

For the prolate ellipsoid (6=c, a> b) we find 

2(l-e 2 )/,. l+e \ .... 



1 i_ e 2 1 + e 
*>-*-?- ^r lo gfT- e ' ( 14 ) 

where e is the eccentricity of the meridian section. The formulae for an oblate ellipsoid 
are given in Art. 373. The values of k for a prolate ellipsoid moving respectively ' end-on 5 
and ' broadside on,' viz. 

*'=ipv *-A' (15) 

are tabulated on the opposite page for a series of values of the ratio a/6. 

For an elliptic disk (a-*-0) the formula (11) becomes nugatory, since a -^2. A separate 
calculation, starting from (1) and (3), leads to the result 

2T=§7rpb 2 c*U*+ P' J(b* sin* 6 + c* cos? 0)d6 (16) 

For b = c this reproduces the result (20) of Art. 102. 

115. We next inquire whether the equation V 2 </> = can be satisfied by 

4> = y*x> C 1 ) 

where % is a function of X only. This requires 

^!+!!h «> 

Now, from Art. 112 (4), (6), 



2 3% + 23x = 2/t2 /19£ + ia ? \dx 
ydy z dz x \y dX z dXJ dX 



ydy z dz \y 

(a 2 + \)(6 2 + X)(c 2 + X) / 1 1 \d X 

(\-fi)(\-v) \& 2 +\ c 2 + XjdX 

On substitution in (2) we find, by Art. 112 (7), 



whence 



X ~ C )k (& a + X)(c 2 +\)A' (3) 



the second constant of integration being chosen as before. 

For a rigid ellipsoid rotating about the axis of x with angular velocity 
£l x> the surface-condition is 

£-*('!->5). ^ 

for X == 0. Assuming * 

*-cy>i (i > + x)^ + x)A (5) 

* The expression (5) differs only by a factor from 

fa 9<I> 

where $ is the gravitation potential of a uniform solid ellipsoid at an external point (x, y, z). 
Since V 2 $ = it easily follows that the above is also a solution of the equation V 2 = O. 



114-115] 



Rotation of an Ellipsoid 



155 



we find that the surface-condition (4) is satisfied, provided 
G 



or 



<7 = 



* ° fe + cV abc\b* -°c 2 ) " *°* (p ~ ?) 
(6 2 -c 2 ) 2 



a^cOa; (6) 



2(6 2 -c 2 ) + (6 2 + c 2 )(/3o- 7 o) 
The formulae for the cases of rotation about 3/ or z can be written down from 
symmetry*. 

The formula for the kinetic energy is 

if (Z, m, n) denote the direction-cosines of the normal to the ellipsoid. The 
latter integral 

=///(2/ 2 - *) dxdydz = J (6 2 - c 2 ) . J7ra6c. 
Hence we find 

2T-1 (6 2 -C 2 ) 2 (70-^0) 4— .fc-'n 2 /«7X 

For a prolate ellipsoid (6 = c, a>6) rotating about an equatorial diameter, the ratio of 
the inertia coefficient to the moment of inertia, about the same diameter, of the mass of 
fluid displaced is found to be 



e 4 (/3 -«o) 



(2-e 2 ){2 e *-(2-* 2 )(A)-«o)} 

The values of k u k 2 (defined in Art. 114), and Id are shewn in the accompanying table 



(8) 



a\b 


*i 


fco 


k' 


1 


0-5 


0-5 





1-50 


0-305 


0-621 


0-094 


2-00 


0-209 


0-702 


0-240 


2-51 


0-156 


0-763 


0-367 


2-99 


0-122 


0-803 


0-465 


3-99 


0-082 


0-860 


0-608 


4-99 


0-059 


0-895 


0-701 


6-01 


0-045 


0-918 


0-764 


6-97 


0-036 


0-933 


0-805 


8-01 


0-029 


0-945 


0-840 


9-02 


0-024 


0-954 


0-865 


9-97 


0-021 


0-960 


0-883 


00 





1 


1 



The two remaining types of ellipsoidal harmonic of the second order, finite at the 
origin, are given by the expression 

*2 „2 ,2 



a 2 + 



tf + + c 2 + 6 



1, 



•(9) 



* The solution contained in (5) and (6) is due to Clebsch, "Ueber die Bewegung einee 
Ellipsoides in einer tropfbaren Flussigkeit, " Crelle, Hi. 103, liii. 287 (1854-6). 



156 Irrotational Motion of a Liquid [chap, v 

where 6 is either root of d^3 + W+6 + c T +6 = ' ^ 10) 

this being the condition that (9) should satisfy V 2 <£ = 0. 

The method of obtaining the corresponding solutions for the external space is explained 
in the treatise of Ferrers. These solutions would enable us to express the motion produced 
in a surrounding liquid by variations in the lengths of the axes of an ellipsoid, subject to 
the condition of no variation of volume : 

d/a + b/b + c/e=0 (11) 

We have already found in Art. 113, the solution for the case where the ellipsoid expands 
(or contracts) remaining similar to itself; so that by superposition we could obtain the 
case of an internal boundary changing its position and dimensions in any manner what- 
ever, subject only to the condition of remaining ellipsoidal. This extension of the results 
arrived at by Green and Clebsch was first treated, though in a different manner from 
that here indicated, by Bjerknes*. 

116. The investigations of this chapter have related almost entirely to 
the case of spherical or ellipsoidal boundaries. It will be understood that 
solutions of the equation V 2 ^> = can be carried out, on lines more or less 
similar, which are appropriate to other forms of boundary. The surface 
which comes next in interest, from the point of view of the present subject, 
is that of the anchor-ring or 'torus'; this case has been very ably treated, by 
distinct methods, by Hicks, and Dyson f. We may also refer to the analyti- 
cally remarkable problem of the spherical bowl, which has been investigated 
by Basset J. 



APPENDIX TO CHAPTER V 

THE HYDRODYNAMICAL EQUATIONS REFERRED TO 
GENERAL ORTHOGONAL CO-ORDINATES 

"We follow the notation of Art. Ill, with this modification that differentiations of 
#, y, z with respect to the independent variables X, /*, v are indicated by the suffixes 1, 2, 3, 
respectively. Thus the direction-cosines of the normal to the surface X = const, are 

{h x x u h x y u Ai^i), 
and so on. 

If u, v, w be the component velocities along the three normals, the total flux out of the 
quasi-rectangular region whose edges are SX/Ax, 8/x/A 2 , &vjh 3 will be 

»(« a+ '(^) V+ »(^!) fc , 

ex \ h 2 n 3 J dfj.\ h z /i x J dv\ h x ti 2 J 

whence the expression for the expansion, viz. 

A-AtM, g (JL) + 1 (^) +| (^)} ; (i) 

cf. Art. Ill (4). 

* I.e. ante p. 147. 

f Hicks. "On Toroidal Functions," Phil. Trans, clxxii. 609 (1881); Dyson, "On the Potential 
of an Anchor-Ring," Phil. Trans, clxxxiv. 43 (1892); see also C. Neumann, I.e. ante p. 134. 

J "On the Potential of an Electrified Spherical Bowl, &c," Proc. Lond. Math. Soc. (1) xv* 
286 (1885) ; Hydrodynamics, i. 149. 



ii5-ii6] Orthogonal Co-ordinates 157 

The circulation round a rectangular circuit on the surface X = const., whose sides are 
§/i/A 2 , 8v/h 3 , is 

i(th-im»* « 

Dividing by the area of the circuit we get the first of the following formulae for the com- 
ponents of vorticity about the three normals : 

«-**{£©-£©}■) 



,A, {sL©"c®r 



To find expressions for the component accelerations, we note that in a time 8t a particle 
changes its parameters from (X, n, v) to (X + SX, /x + fy*, v + bv\ where 

8k/ h = u tit, 8[xlh 2 = vdt, 8v/h 3 =w8t. 

The component velocities therefore become 



U + \ ^+^l w 5T + ^2 v 5 _ + ^3^ ; T-)^ &C-, & c -j 



•(4) 



and we have to resolve these along the original directions of u, v, w. Now after a time 8t 
the direction-cosines of the new direction of v become 



h 2 x 2 + ^ (^2^2) hiudt + ^- (h 2 x 2 )k 2 v8t + —_ (h 2 x 2 ) h 3 w8t, &c, &c, 



75 7) ?) 

^- (h 2 x 2 ) hiubt + 7r- (h 2 x 2 )k 2 v8t + — 

Ck " OfX cv 

where in the two expressions not written out the derivatives of x are to be replaced by 
those of y and z, respectively. Hence the cosine of the angle between the new direction of 
v and the original direction of u, viz. (A^, Ai#i, Ai^i), is 

{{x x x 12 +y x y l2 + z x z 12 ) h x u + (x 1 x 22 + y x y 22 + z x z 22 ) h 2 v + fa x 23 +y t y 2 3 + *i ^23) h w) h x h 2 8t. 

(5) 

Certain terms have been omitted from this expression in virtue of the relation 

•^1^2+^1^2 + ^1^2 = 0, (6) 

which follows from the orthogonal property. Again, differentiating (6) with respect to v 
and comparing with similar results we infer that 

#1 #23+3^1 #23 + *1*23 = (7)* 

Also, differentiation of the identity 

^2 +yi 2 + , i2== _L (8) 

with respect to p gives #i#i2 + ? /i#]2+zi 2 i2=T- ~ (t) W 

hi Cfi \tiiJ 

Again, 

^1^22+yiy22 + 2l222 = g-(^1^2+yi3/2 + 2l22)-(^2^12 + 2/iyi2 + ^l2l2)= ~J ^ (jY .-.(10) 

The expression (5) thus reduces to 

{4,d)-4Xi)} wt (n) 

* Forsyth, Differential Geometry, Cambridge (1912), p. 412. 



158 Orthogonal Co-ordinates [chap, v 

In the same way the cosine of the angle between the new direction of w and the 
original direction of u is 



{4(0-4®} m* 



.(12) 



The acceleration in the original direction of u is thus found to be 
du t , du , , du t , du 



or, more symmetrically, 

du 
dt l " l ""W , ' riV d l i l ' v ' sw di 



+A ^4(£)-4(£)} 



.(13)* 



37 + A 1 w^ r +A 2 v — +A 3 w 



.*, {^iO'+^ts+M-i®} m 

The expressions for the acceleration in the direction of v and w follow by symmetry. 

For example, in cylindrical co-ordinates we have 

#=rcos#, # y = rsin0, z—z. 
Putting X = r, p = 0, v=z, 

we have A x = 1, A 2 = l/ r > A 3 =l. 

The expansion is accordingly 



du u dv dw 
^Tr+r+We + lz' (15) 



and the components of vorticity are 





. dw dv 

s~r~dd~dz~> 


v- 


du dw 
dz dr 


, C= 


dv 


The 


component accelerations are 












du 

Tt +U 


du 
dr 


du 


v 2 du \ 

- +w y , 
r oz 




8» 


dv 

dr 


dv 
rdd 


uv , dv 
—+w^, 

r dz 




dw 

Tt +U 


dw 
dr 


dw dw 
rdd oz 


J 



V ou 
r rd9 



.(16) 



.(17) 



If in this formula we put w=0 we get the results for plane polar co-ordinates (Art. 16 a). 
In spherical polar s 

x=r sin 0coso>, y = r sin 6 sin o>, 2=rcos#. 
Putting X = r, fi = 0, v = a>, 

we have h x = 1, h 2 = 1/r, h 3 = 1/r sin 6. 

* G. B. Jeffery, Phil. Mag. (6) xxix. 445 (1915). 



APP. 



Orthogonal Co-ordinates 



159 



Hence 



A du u dv v 1 dw 

A = 3- + 2- +-57. + -cot#H r—. >. ^-, 

3* r roB r r sin 6 do 



.(18) 



*-=£- 



3-y 



rd& rsmOda* r 



w 

— cot 0, 



ou 



dw 



r sin 6d<o dr r 



.(19) 






The component accelerations are 

3m 3m 

+m ^- + v 



3m v 2 + w 2 



3* 3r rd& rsinBdo 

dv 



dv dv dv 



uv w* a 
ot or rdO rsmBdo r r 



•(20) 



dw dw dw dw 

W^ U Yr +V rle^ W r sin ddo^^ 



WW vw , . 

+ — + — cot 6 ; 
r 



cf. Art. 16 a. 



CHAPTER VI 

ON THE MOTION OF SOLIDS THROUGH A LIQUID: 
DYNAMICAL THEORY 

117. In this chapter it is proposed to study the very interesting 
dynamical problem furnished by the motion of one or more solids in a 
frictionless liquid. The development of this subject is due mainly to 
Thomson and Tait* and to Kirchhofff. The cardinal feature of the methods 
followed by these writers consists in this, that the solids and the fluid are 
treated as forming together one dynamical system, and thus the troublesome 
calculation of the effect of the fluid pressures on the surfaces of the solids is 
avoided. 

To begin with the case of a single solid moving through an infinite mass 
of liquid, we will suppose in the first instance that the motion of the fluid is 
entirely due to that of the solid, and is therefore irrotational and acyclic. 
Some special cases of this problem have been treated incidentally in the 
foregoing pages, and it appeared that the whole effect of the fluid might be 
represented by an addition to the inertia of the solid. The same result will 
be found to hold in general, provided we use the term 'inertia' in a somewhat 
extended sense. 

Under the circumstances supposed, tne motion of the fluid is characterized 
by the existence of a single- valued velocity-potential (j> which, besides satis- 
fying the equation of continuity 

V 2 <£ = 0, (1) 

fulfils the following conditions : (1°) the value of —d<f>/dn, where $n denotes 
as usual an element of the normal at any point of the surface of the solid, 
drawn on the side of the fluid, must be equal to the velocity of the surface 
at that point normal to itself, and (2°) the differential coefficients dcf>/da), 
d4>/dy, d(f>/dz must vanish at an infinite distance, in every direction, from the 
solid. The latter condition is rendered necessary by the consideration that 
a finite velocity at infinity would imply an infinite kinetic energy, which 
could not be generated by finite forces acting for a finite time on the solid. 
It is also the condition to which we are led by supposing the fluid to be 
enclosed within a fixed vessel infinitely large and infinitely distant, all round, 
from the moving body. For on this supposition the space occupied by the 
fluid may be conceived as made up of tubes of flow which begin and end on 

* Natural Philosophy, Art. 320. Subsequent investigations by Lord Kelvin will be referred 
to later. 

t "Ueber die Bewegung eines Kotationskorpers in einer Fliissigkeit," Crelle, lxxi. 237 (1869) 
[Ges. Abh. p. 376]; Mechanik, c. xix. 



H7-H9] Impulse of the Motion 161 

the surface of the solid, so that the total flux across any area, finite or 
infinite, drawn in the fluid must be finite, and therefore the velocity at 
infinity zero. 

It has been shewn in Art. 41 that under the above conditions the motion 
of the fluid is determinate. 

118. In the further study of the problem it is convenient to follow the 
method introduced by Euler in the dynamics of rigid bodies, and to adopt a 
system of rectangular axes Ox, Oy, Oz fixed in the body, and moving with it. 
If the motion of the body at any instant be defined by the angular velocities 
p, q, r about, and the translational velocities u, v, w of the origin parallel to, 
the instantaneous positions of these axes *, we may write, after Kirchhoff, 

^ = ^l + V^2 + ^3+^%l + g , %2 + ^%3, (2) 

where, as will appear immediately, <j> ly <£ 2 , </>3> %i> %2> %3 are certain functions 
of x, y, z determined solely by the configuration of the surface of the solid, 
relative to the co-ordinate axes. In fact, if I, m, n denote the direction-cosines 
of the normal, drawn towards the fluid, at any point of this surface, the 
kinematical surface-condition is 

rich 

— ^— = l(u + qz — ry) + m (v + rx — pz) + n(w+py — qx\ 
whence, substituting the value (2) of <£, we find 

-^ =n ) 

(3) 



9 *1 _/ ^2 _ 903 

r — I, ~ — fit, ~ — n, 

on on on 



9%1 C '%2 7 3^3 7 

— £- = ny — mz, — -^- = iz — nx, — -^- = mx — I 1 
dn * dn dn 

Since these functions must also satisfy (1), and have their derivatives zero at 
infinity, they are completely determinate, by Art. 41 f. 

119. Now whatever the motion of the solid and fluid at any instant, it 
might have been generated instantaneously from rest by a properly adjusted 
impulsive ' wrench ' applied to the solid. This wrench is in fact that which 
would be required to counteract the impulsive pressures pcf> on the surface, 
and, in addition, to generate the actual momentum of the solid. It is called 
by Lord Kelvin the ' impulse ' of the system at the moment under con- 
sideration. It is to be noted that the impulse, as thus defined, cannot be 
asserted to be equivalent to the total momentum of the system, which is 
indeed in the present problem indeterminate J. We proceed to shew however 
that the impulse varies, in consequence of extraneous forces acting on the 
solid, in exactly the same way as the momentum of a finite dynamical system. 

* The symbols u, v, w, p, q, r are not at present required in their former meanings, 
t For the particular ease of an ellipsoidal surface, their values may be written down from 
the results of Arts. 114, 115. 

J That is, the attempt to calculate it leads to 'improper' or 'indeterminate' integrals. 



162 Motion of Solids through a Liquid [chap, vi 

Let us in the first instance consider any actual motion of a solid, from 
time t to time t ly under any given forces applied to it, in a finite mass 
of liquid enclosed by a fixed envelope of any form. Let us imagine the 
motion to have been generated from rest, previously to the time t , by forces 
(whether continuous or impulsive) applied to the solid, and to be arrested, in 
like manner, by forces applied to the solid after the time fa. Since the 
momentum of the system is null both at the beginning and at the end of this 
process, the time-integrals of the forces applied to the solid, together with 
the time-integral of the pressures exerted on the fluid by the envelope, must 
form an equilibrating system. The effect of these latter pressures may be 
calculated, by Art. 20, from the formula 

J-g-w+'w a> 

A pressure uniform over the envelope has no resultant effect ; hence, since <j> 
is constant at the beginning and end, the only effective part of the integral 
pressure fp dt is given by the term 

-iplq'dt (2) 

Let us now revert to the original form of our problem, and suppose the 
containing envelope to be infinitely large, and infinitely distant in every 
direction from the moving solid. It is easily seen by considering the arrange- 
ment of the tubes of flow (Art. 36) that the fluid velocity q at a great 
distance r from an origin in the neighbourhood of the solid will ultimately 
be, at most*, of the order 1/r 2 , and the integral pressure (2) therefore of the 
order 1/r 4 . Since the surface-elements of the envelope are of the order r^-sr, 
where Scr is an elementary solid angle, the force- and couple-resultants of the 
integral pressure (2) will now both be null. The same statement therefore 
holds with regard to the time-integral of the forces applied to the solid. 

If we imagine the motion to have been started instantaneously at time t , 
and to be arrested instantaneously at time t 1} the result at which we have 
arrived may be stated as follows : 

The ' impulse ' of the motion (in Lord Kelvin's sense) at time t± differs 
from the ' impulse ' at time t by the time-integral of the extraneous forces 
acting on the solid during the interval t\ — t f. 

It will be noticed that the above reasoning is substantially unaltered 
when the single solid is replaced by a group of solids, which may moreover 
be flexible instead of rigid, and even when these solids are replaced by 
masses of liquid which are moving rotationally. 

120. To express the above result analytically, let f, rj, f, \, /jl, v be the 
components of the force- and couple-constituents of the impulse ; and let 

* It is really of the order 1/r 3 when, as in the case considered, the total flux outwards is zero, 
t Sir W. Thomson, I.e. ante p. 33. The form of the argument given above was kindly- 
suggested to the author by Sir J. Larmor. 



ii9-i2i] Kinetic Energy 163 

X, Y, Z, L, M, N designate in the same manner the system of extraneous 
forces. The whole variation of £, rj, J, X, yu., v, due partly to the motion of the 
axes to which these quantities are referred, and partly to the action of the 
extraneous forces, is then given by the formulae* 

-i=r/7-g?+X, ^- = wrj - v£ + rp- qv + Z, ^ 

g = p f - r £ + F, d £=uZ-wZ+pp-r\ + M,\ (1) 

TO 7 

-± = q£- pv + Z, ~ = v£-u V J-q\-pfx + N. 

For at time t + 8t the moving axes make with their positions at time t 
angles whose cosines are 

(1, r8t, -qSt), (-r8t, 1, p8t), (q8t, -p8t, 1), 
respectively. Hence, resolving parallel to the new position of the axis of x y 
£ + 8f;=!; + v.rSt-Z.q$t + X8t. 

Again, taking moments about the new position of Ox, and remembering that 
has been displaced through spaces u8t, v8t, w8t parallel to the axes, we find 

\ + 8\ = \ + r).w8t — %.vht + /j..r8t — v .q8t + L8t. 

These, with the similar results which can be written down from symmetry, 
give the equations (1). 

When no extraneous forces act, we verify at once that these equations 
have the integrals 

f a + V 2 + ? 2 = const., \% + m + v£= const., (2) 

which express that the magnitudes of the force- and couple-resultants of the 
impulse are constant. 

121. It remains to express f, rj, f, \, /jl, v in terms of u, v, w, p, q, r. In 
the first place let T denote the kinetic energy of the fluid, so that 



2T 



->li*£* « 



where the integration extends over the surface of the moving solid. Substi- 
tuting the value of </> from Art. 118 (2), we get 

2T = Aw 2 + Bv 2 + Cw 2 + 2A'vw + 2B' wu + 2C'uv 

+ Tp 2 + Qq 2 + Rr 2 + 2T'qr + 2Q' rp + 2R'pq 

+ 2p (Fu + Gv + Hw) + 2q (F'u + G'v + U'w) + 2r (F"u + G"v + H"w), 

(2) 

where the twenty-one coefficients A, B, C, &c. are certain constants 

* Cf. Hayward, " On a Direct Method of Estimating Velocities, Accelerations, and all similar 
Quantities, with respect to Axes moveable in any manner in space," Camb. Trans, x. 1 (1856). 



164 



Motion of Solids through a Liquid [chap, vi 



determined by the form and position of the surface relative to the co-ordinate 
axes. Thus, for example, 



A 
A 






\ 



-■-*JJ*£*~'JJ*&" 



•(3) 



= p \\ fan dS — p <j> z m dS, 

* — P //» ^ dS = P jjxi (Jiy - mz) dS, i 

the transformations depending on Art. 118 (3) and on a particular case of 
Green's Theorem (Art. 44 (2)). These expressions for the coefficients were 
given by Kirch hofT. 

The actual values of the coefficients in the expression for 2T have been found in the 
preceding chapter for the case of the ellipsoid, viz. we have from Arts. 114, 115 



_«c 
2-a 



A = ^- .^npabc, 



(& 2 -c 2 ) 2 (yo-/3 ) 

5 2(&2- C 2) + (62 + c 2 )(/3() _ 7o ) 



. I npabc, 



•(4) 



with similar expressions for B, 0, Q, R. The remaining coefficients, as will appear pre- 
sently, in this case all vanish. We note that 

2(a -/3o) 



A-B= 



. ^ npabc, 



.(5) 



(2-« )(2-/3 ) 

so that if a > b > c, then A< B < O, as might have been anticipated. 

The formulae for an ellipsoid of revolution may be deduced by putting b = c ; they may 
also be obtained independently by the method of Arts. 104-109. Thus for a circular disk 
(a=0, b = c) we have 

A, B, C = f pC 3, 0, 0; P, Q, R = 0, l%pc\ ^P<* (6) 

121 a. When the motion of the solid is one of pure translation the formula 
for the kinetic energy of the fluid reduces to 

2T = Au 2 + Bv 2 +Cw 2 + 2A'vw + 2B'wu + 2C'uv (1) 

We can now shew that the effect at a great distance is in all cases that of a 
suitable double source, and that the character of this source is completely 
defined by the coefficients in (1). 

For this we have recourse to the formula (12) of Art. 58, viz. 

,J 1 



Wp-Jjtf-^-dS. 



■(2) 



We may regard the boundary of the solid as a thin rigid shell, with fluid also 
in its interior, and assume the potentials </> and cp' to refer to the external 



121-121 a] Effect at a Distance 165 

and internal regions, respectively. Let (x 1} y 1} z-±) be the co-ordinates of the 
point P, which we suppose to be at a distance great compared with the 
dimensions of the solid, and (x, y, z) those of a surface-element SS. Then, 
writing 

n = V(% 2 + 2/! 2 + z?\ r = V{(^i - xf + (y x - yf + {z x - zf}, 
we have, approximately, 

1 _ 1 xx-i + yy x + zz-i 3 1_ lx\ + my x 4- nz x 
r Tx ^i 3 ' dn r rj? 

Suppose, now, that the shell is moving with unit velocity parallel to x, 
without rotation. Writing 

= 1? f = -*, (3) 

u {{j. dl jv Ax 1 + C'y 1 + B'z 1 ... 

we have \\6 = dS = — ^ , (4) 

JJ^dnr prf 

and JJ* r s>~£- < 5 > 

where Q denotes the volume of the solid. We have, in fact, 

llwldS=Q, ljxmdS = 0, llxndS = (6) 

Hence ^ ^ p JA±PQ^l±pl±^3 (7) * 

The effect at a distance is therefore that of a double source, but the axis 
of the source does not necessarily coincide with the direction of transition. 
If, however, the solid is moving parallel to an axis of permanent translation 
(Art. 124), the coefficients C and B' vanish, and 

4^ = <A±^ (8) 

For example, in the case of the sphere we have A = §7r/oa 3 , Q = |7ra 3 , and 

^ = 2^' (9) 

as in Art. 92. 

When the velocity (u, v, w) of the solid is general, the formula (7) is 
replaced by 

47rr 1 3 / o0p = (Au + C'v + B'w) x x 

+ (G'u + Bv + A'w) y x \ {B'u + A'v + Cw) z x 

+ pQ(ux 1 + vy 1 + wz-s) (10) 

Conversely a knowledge of the form of the velocity-potential at infinity due to a ' per- 
manent ' translation leads to a knowledge of the corresponding inertia-coefficient. 

* From a paper "On Wave Eesistance," Proc. Roy. Soc. cxi. 15 (1926). 



166 Motion of Solids through a Liquid [chap, vi 

For instance, in the Rankine ovals referred to in Art. 97 we have a distribution of 
sources along the axis of x, subject to the condition that the total 'strength' of these 
sources is zero. If the line-density of this distribution be m, we have 

V J >/{{*! - 1) 2 +yi 2 + *i 2 } J Vi rf J g ' 
or #-^ J **£<% + ..-, (11) 

since jmd£=0. Hence 

A/p + Q=47rSm£d£ (12)* 

122. The kinetic energy, Ti say, of the solid alone is given by an expres- 
sion of the form 

+ Pii> 2 + Qiq 2 + Rir 2 -r 2PiV + 2Qi'rp + 2R 1 'pq 

+ 2m {a (vr — wq) + /? (wp — ur) + y (uq — vp)} (1) 

Hence the total energy T+-Ti, of the system, which we shall denote by T, is 
given by an expression of the same general form as in Art. 121, say 

2T = An 2 + Bv 2 + Gw 2 + 2A'vw + 2B'wu + 2C'uv 
+ Pp 2 + Qq 2 + Rr 2 + 2Fqr + 2Q'rp + 2R'pq 
+ 2p (Fu + Gv + Hw) + 2q (F'u + G'v + H'w) + 2r {F"u + G"v + H "w). 

(2) 

The values of the several components of the impulse in terms of the velo- 
cities u, v, w, p, q, r can now be found by a well-known dynamical method f. 
Let a system of indefinitely great forces (X, Y, Z, L, M, N) act for an 
indefinitely short time t on the solid, so as to change the impulse from 
(ft 17, f, \, fi, v) to (? + S£ v + H ?+8£ \ + B\, /i + S/x, * + &/). The work 
done by the force X, viz. 

\ T Xudt, 

J o 

lies between % X cfa and u 2 X dt, 

Jo Jo 

where ui and u 2 are the greatest and least values of u during the time t, 
i.e. it lies between %8f and u 2 B^. If we now introduce the supposition that 
Sf, S77, Sf, 8\, $/z, 81/ are infinitely small, % and u 2 are each equal to u, and 
the work done is uB%. In the same way we may calculate the work done by 
the remaining forces and couples. The total result must be equal to the 
increment of the kinetic energy, whence 

u 8f + v Brj + w 8f + p BX + q Bfi + r Bv 

= BT^Su + d ^8v + ^Sw + d ^Bp + fs q + d ^Sr. ...(3) 
ou ov ow dp * oq or 

* G. I. Taylor, Proc. Roy. Soc, cxx. 13 (1928). 

t See Thomson and Tait, Art. 313, or Maxwell, Electricity and Magnetism, Part iv. c. v. 



i2ia-i22] Relations between Energy and Impulse 167 

Now if the velocities be all altered in any given ratio, the impulses will 
be altered in the same ratio. If then we take 



8u 8v 
u V 


8w 8p 
w p 


_ B 2 _ Br _ k 

q r 


it will follow that S£=h = 


8£ 8\ 


hfj, 8v , 

/J, V 


Substituting in (3), we find 






u!; + vt) + w%+p\ + qp + rv 







dT dT dT dT , dT dT om ... 
= u ^ + v ^ + w ^ + P^ + lY q +r ^ =2f ' -< 4 > 

since T is a homogeneous quadratic function. Now performing the arbitrary 
variation 8 on the first and last members of (4), and omitting terms which 
cancel by (3), we find 

f$w + v 8v + %8w + \8p + fi8q + v8r = 8T. 

Since the variations 8u, 8v, 8iu, 8p, 8q, 8r are all independent, this gives the 
required formulae 

„ dT dT dT % dT dT dT /KX 

?' * ?= ^' *i> di>> X ' * " = ^' dq> SF (5) 

It may be noted that since jf , 17, J, . . . are linear functions of w, v, w, . . . , 
the latter quantities may also be expressed as linear functions of the former, so 
that T may be regarded as a homogeneous quadratic function of f, 77, f, \, /jl, v. 
When expressed in this manner we may denote it by T\ The equation (3) 
then gives at once 

u 8f + v 8rj + w 8£ + p 8\ + q 8/j, + r 8v 

dT\ t dT\ ^\ y .dT"dT\ dT\ 

dT" dr dT s dr dT dr 

whence „,*, w = __,_, ^, g , r = __,__, __ ( 6 ) 

These formulae are in a sense reciprocal to (5). 

We can utilize this last result to obtain, when no extraneous forces act, 
another integral of the equations of motion, in addition to those found in 
Art. 120. Thus 

dt ~ d% dt + '" + '" + ax dt^'"^'" 

d% dX 

which vanishes identically, by Art. 120 (1). Hence we have the equation of 
energy 

T=const ...(7) 



168 



Motion of Solids through a Liquid [chap, vi 



123. If in the formulae (5) we put, in the notation of Art. 121, 

T=T + T 1 , 

it is known from the Dynamics of rigid bodies that the terms in T x represent 
the linear and angular momentum of the solid by itself. Hence the remaining 
terms, involving T, must represent the system of impulsive pressures exerted 
by the surface of the solid on the fluid, in the supposed instantaneous genera- 
tion of the motion from rest. 

This is easily verified. For example, the ^-component of the above system 
of impulsive pressures is 



= Au + C'v + B'w + Pp + ¥'q + F'V 



8T 



.(8) 



by the formulae of Arts. 118, 121. In the same way, the moment of the 
impulsive pressures about Ox is 

dxi 



))p<t>( n y 



mz) dS 



Fu + Gv + Kw + Pp + H'q + Q V 



*~£' dS 



dr 

dp ' 



(9) 



124. The equations of motion may now be written* 



ddT 

dt du 

ddT 

dt dv 

d dT 



dT dT _ 



dv 
dT 



dw 



\ 



^ dw du ' 



dT dT 



dt dw ^ du P dv 

d dT dT dT dT 

dt dp dv dw dq 



+ Z, 



dT T 
^d~r +L ' 



(1) 



ddT 

dt dq 



dT 

dw 



±dT = dT 

dt dr du 



dT 

du 

dT 



T + ^ = u ^- w *r+P^:- r ^ + M > 



dT 

dr 



dT 



dv ^dp P 



dT 
dp 

dT 



+ N. 



If in these we write T= T + Ti, and isolate the terms due to T, we obtain 
expressions for the forces exerted on the moving solid by the pressure of the 
surrounding fluid; thus the total component (X, say) of the fluid pressure 
parallel to x is 

ddT dT dT 



X = 



dt du, dv dw' 



•(2) 



* See Kirchhoff, I.e. ante p. 160; also Sir W. Thomson, " Hypokinetic Solutions and Obser- 
vations," Phil. Blag. (5) xlii. 362 (1871) [reprinted in Baltimore Lectures, Cambridge, 1904, p. 584]. 



123-124] Equations of Motion 169 

and the moment (L) of the same pressures about x is * 

dt dp dv dw dq ^ dr' 

For example, if the solid be constrained to move with a constant velocity 
(u, v, w\ without rotation, we have 

X, Y, Z - 0, ) 

T nwr ^ ST dT dT dT dT dT \ (4) 

L, M, N = w ^ # — , u^ w x- , Vr u-^- , 

dv dw dw du du dv ) 

where 2T = Au 2 + Bv 2 + Cw 2 + 2A'vw + 2B' wu + 2C'uv. 

The fluid pressures thus reduce to a couple, which moreover vanishes if 

dT = aT _3T 

du' dv ' dw' ' 
i.e. provided the velocity (u, v, w) be in the direction of one of the principal 
axes of the ellipsoid 

Ax 2 + By 2 + Cz 2 + 2A'yz + ZB'zx + 2C'xy = const (5) 

Hence, as was first pointed out by Kirchhoff, there are, for any solid, 
three mutually perpendicular directions of permanent translation; that is 
to say, if the solid be set in motion parallel to one of these, without rotation, 
and left to itself, it will continue to move in this manner. It is evident that 
these directions are determined solely by the configuration of the surface of 
the body. It must be observed however that the impulse necessary to produce 
one of these permanent translations does not in general reduce to a single 
force ; thus if the axes of co-ordinates be chosen, for simplicity, parallel to 
the three directions in question, so that A' t B r , C = 0, we have, corresponding 
to the motion u alone, 

f, v , Z=Au, 0, 0; X, //,, v—Fu, F'u, F"u, 
so that the impulse consists of a wrench of pitch FjA. 

With the same choice of axes, the components of the couple which is the 
equivalent of the fluid pressures on the solid, in the case of any uniform 
translation (u, v, w), are 

L, M, N = (B-C)w;, (C-A)ww, (A-B)w; (6) 

Hence if in the ellipsoid 

Ax 2 + By 2 +Cz 2 = const., (7) 

we draw a radius vector r in the direction of the velocity (u, v, w) and erect 

the perpendicular h from the centre on the tangent plane at the extremity 

of r, the plane of the couple is that of h and r, its magnitude is proportional 

to sin (h, r)/h, and its tendency is to turn the solid in the direction from h to r. 

* The forms of these expressions being known, it is not difficult to verify them by direct 
calculation from the pressure-equation, Art. 20 (5). See a paper "On the Forces experienced by 
a Solid moving through a Liquid," Quart. Journ. Math. xix. 66 (1883). 



170 Motion of Solids through a Liquid [chap, vi 

Thus if the direction of (u, v, w) differs but slightly from that of the axis 
of x, the tendency of the couple is to diminish the deviation when A is the 
greatest, and to increase it when A is the least, of the three quantities A, B, C, 
whilst if A is intermediate to B and C the tendency depends on the position 
of r relative to the circular sections of the above ellipsoid. It appears then 
that of the three permanent translations one only is thoroughly stable, viz. 
that corresponding to the greatest of the three coefficients A, B, C. For 
example, the only stable direction of translation of an ellipsoid is that of its 
least axis; see Art. 121*. 

125. The above, although the simplest, are not the only steady motions 
of which the body is capable, under the action of no extraneous forces. The 
instantaneous motion of the body at any instant consists, by a well-known 
theorem of Kinematics, of a twist about a certain screw ; and the condition 
that this motion should be permanent is that it should not affect the 
configuration of the impulse (which is fixed in space) relatively to the body. 
This requires that the axes of the screw and of the corresponding impulsive 
wrench should coincide. Since the general equations of a straight line involve 
four independent constants, this gives four linear relations to be satisfied by 
the five ratios u : v; w: p : q : r. There exists then for every body, under the 
circumstances here considered, a singly-infinite system of possible steady 
motions. 

The steady motions next in importance to the three permanent translations are those 
in which the impulse reduces to a couple. The equations (1) of Art. 120 shew that we 
may have £, 77, £= 0, and X, p, v constant, provided 

X/p=fM/q= v /r, =k, say (1) 

If the axes of co-ordinates have the special directions referred to in the preceding Art., the 
conditions £, 77, £=0 give us at once u, v, w in terms of p, q, r, viz. 

Fp + F'q + F"r Gp + G'q + G"r Hp + H'q+H"r 

u ~ A ' V ~ B ' W C () 

Substituting these values in the expressions for X, /u, v obtained from Art. 122 (5), we find 

de de de 
*>» v =ty> ty> «■' < 3 > 

provided 29 (p, q, r) = typ 2 + i&q 2 + %ir 2 + 2Wqr + 2i$rp+2Wpq, (4) 

the coefficients in this expression being determined by formulae of the types 

m-P-^1-^-^ m P> *"*"' Q ' G " H ' H " ^ 

** A B C ®~ A B C W 

These formulae hold for any case in which the force-constituent of the impulse is zero. 
Introducing the conditions (1) of steady motion, the ratios p : q : r are to be determined 
from the three equations 

\Bp + &q + <B'r = kp,^ 

Wp + ®q + W'r=kq\ (6) 

<&p + '$q+1&r = kr. J 

* The physical cause of this tendency of an elongated body to set itself broadside-on to the 
relative motion is clearly indicated in the diagram on p. 86. A number of interesting practical 
illustrations are given by Thomson and Tait, Art. 325. 



124-125] Steady Motions 171 

The form of these shews that the line whose direction-ratios are p:q :r must be parallel 
to one of the principal axes of the ellipsoid 

Q(x, y, 2) = const (7) 

There are therefore three permanent screw-motions such that the corresponding impulsive 
wrench in each case reduces to a couple only. The axes of these three screws are mutually 
at right angles, but do not in general intersect. 

It may now be shewn that in all cases where the impulse reduces to a couple only, the 
motion can be completely determined. It is convenient, retaining the same directions of 
the axes, to change the origin. Now the origin may be transferred to any point (#, y, z) 
by writing 

u + ry-qz, v + pz — rx, w + qx-py, 

for u, v, w respectively. The coefficient of 2vr in the expression for the kinetic energy, 
Art. 122 (2), becomes -Bx+G", that of 2wq becomes Cx + H\ and so on. Hence if we 
take 

X (G" E'\ ,(H F"\ (F' 0\ ... 

the coefficients in the transformed expression for 2T will satisfy the relations 

<r_#' h__f^_ F l_ ( * 

B~C' C~A' A~ B (y) 

If we denote the values of these pairs of equal quantities by a, /3, y respectively, the 
formulae (2) may be written 

d* d* dV ,„,_. 

M= -^' v= ~dj' w= ~w (10) 

where 2* ( p, q, r) =s — p 2 + -^ q 2 -f — ^ r 2 + 2aqr + 2ftrp + 2ypq (11) 

The motion of the body at any instant may be conceived as made up of two parts ; 
viz. a motion of translation equal to that of the origin, and one of rotation about an 
instantaneous axis passing through the origin. Since £, 77, (=0 the latter part is to be 
determined by the equations 

dX du, . dv 

T =r v .-qy i f t =pv-r\ g -flX-W 

which express that the vector (X, /x, v) is constant in magnitude and has a fixed direction 
in space. Substituting from (3), 

d L de_ 9e_ de \ 

dt dp~ dq ^ dr ' 

d de de de 

dtTq=VTr~ r irp^ ' {l " } 

d_de_ de_ de 

dt dr~^dp ™dq 

These are identical in form with the equations of motion of a rigid body about a fixed 
point, so that we may make use of Poinsot's well-known solution of the latter problem. 
The angular motion of the body is obtained by making the ellipsoid (7), which is fixed in 
the body, roll on a plane 

\x + fxy + vz = const. , 

which is fixed in space, with an angular velocity proportional to the length 01 of the 
radius vector drawn from the origin to the point of contact 7. The representation of the 
actual motion is then completed by impressing on the whole system of rolling ellipsoid 



172 Motion of Solids through a Liquid [chap, vi 

and plane a velocity of translation whose components are given by (10). This velocity is 
in the direction of the normal M to the tangent plane of the quadric 

*tay,#)-T-«», (13) 

at the point P where 01 meets it, and is equal to 

<? 3 
np DM xam ? u l ar velocity of body (14) 

When 0/does not meet the quadric (13), but the conjugate quadric obtained by changing 
the sign of e, the sense of the velocity (14) is reversed*. 

126. The problem of the integration of the equations of motion of a solid 
in the general case has engaged the attention of several mathematicians, but, 
as might be anticipated from the complexity of the question, the physical 
meaning of the results is not easily grasped f. 

In what follows we shall in the first place inquire what simplifications 
occur in the formula for the kinetic energy, for special classes of solids, and 
then proceed to investigate one or two particular problems of considerable 
interest which can be treated without difficult mathematics. 

The general expression for the kinetic energy contains, as Ave have seen, 
twenty-one coefficients, but by the choice of special directions for the 
co-ordinate axes, and a special origin, these can be reduced to fifteen J. 
The most symmetrical way of writing the general expression is 
2T = An 2 + Bv 2 + Cw 2 + 2A'vto + 2B'wu + 2C'uv 
+ Pp 2 + Qq 2 + Rr 2 + 2P'qr + 2Q'rp + 2R'pq 
+ 2Lup + 2Mvq + 2Nwr 
+ 2F(vr + wq) + 2G (wp + ur) + 2H(uq + vp) 

+ 2F' (vr-wq)+2G' (wp-ur) + 2H' (uq-vp) (1) 

It has been seen that we may choose the directions of the axes so that 
A\ B' ', C = 0, and it may easily be verified that by displacing the origin 
we can further make F\ G' } H' = 0. We shall henceforward suppose these 
simplifications to have been made. 

1°. If the solid has a plane of symmetry, it is evident from the con- 
figuration of the relative stream-lines that a translation normal to this plane 
must be one of the permanent translations of Art. 124. If we take this plane 
as that of xy y it is further evident that the energy of the motion must be 
unaltered if we reverse the signs of w, p, q. This requires that P', Q', L, M, 
N, H should vanish. The three screws of Art. 125 are now pure rotations, 
but their axes do not in general intersect. 

* The substance of this Art. is taken from a paper, "On the Free Motion of a Solid through 
an Infinite Mass of Liquid," Proc. Lond. Math. Soc. viii. 273 (1877). Similar results were obtained 
independently by Craig,." The Motion of a Solid in a Fluid," Amer. Journ. of Math. ii. 162 (1879). 

f For references see Wien, Lehrbuch d. Hydrodynamik, Leipzig, 1900, p. 164. 

J Cf. Clebsch, "Ueber die Bewegung eines Korpers in einer Flussigkeit," Math. Ann. hi. 
238 (1870). This paper deals with the 'reciprocal' form of the dynamical equations, obtained 
by substituting from Art. 122 (6) in Art. 120 (1). 



125-126] HydroMnetic Symmetries 173 

2°. If the body has a second plane of symmetry, at right angles to the 
former one, we may take this as the plane xz. We find that in this case 
R' and G must also vanish, so that 

2T = Au 2 + Bv 2 +Cw 2 + Pp 2 + Qq 2 + Rr 2 + 2F(vr + wq) (2) 

The axis of x is the axis of one of the permanent rotations, and those of the 
other two intersect it at right angles, though not necessarily in the same point. 

3°. If the body has a third plane of symmetry, say that of yz, at right 
angles to the two former ones, we have 

2T = Au 2 + Bv 2 + Gw 2 + Pp 2 +Qq 2 + Rr 2 (3) 

4°. Returning to (2°), we note that in the case of a solid of revolution 
about Ox, the expression for 2T must be unaltered when we write v,q, — w, — r 
for w, r, v, q, respectively, since this is equivalent to rotating the axes of y, z 
through a right angle. Hence B= G, Q = R, F=0; and therefore 

2T = Au 2 + B(v 2 + w 2 ) + Pp 2 + Q(q 2 + r 2 ) (4)* 

The same reduction obtains in some other cases, for example when the 
solid is a right prism whose section is any regular polygon f. This is seen at 
once from the consideration that, the axis of x coinciding with the axis of the 
prism, it is impossible to assign any uniquely symmetrical directions to the 
axes of y and z. 

5°. If, in the last case, the form of the solid be similarly related to each 
of the co-ordinate planes (for example a sphere, or a cube), the expression (3) 
takes the form 

2T = A(u 2 +v 2 + w 2 ) + P(p 2 + q 2 + r 2 ) (5) 

This again may be extended, for a like reason, to other cases, for example 
any regular polyhedron. Such a body is practically for the present purpose 
'isotropic,' and its motion will be exactly that of a sphere under similar 
conditions. 

6°. We may next consider another class of cases. Let us suppose that 
the body has a sort of skew symmetry about a certain axis (say that of x), 
viz. that it is identical with itself turned through two right angles about this 
axis, but has not necessarily a plane of symmetry]:. The expression for 2T 
must be unaltered when we change the signs of v, w, q, r, so that the 
coefficients Q\ R', G, H must all vanish. We have then 

2T = An 2 + Bv 2 + Gw 2 + Pp 2 + Qq 2 + Rr 2 + 2P'qr 

+ 2Lup + 2Mvq + 2Nwr 4- 2F (vr + wq) (6) 

* For the solution of the equations of motion in this case see Greenhill, "The Motion of a 
Solid in Infinite Liquid under no Forces," Amer. Journ. of Math. xx. 1 (1897). 

t SeeLarmor, "On Hydrokinetic Symmetry," Quart. Journ. Math. xx. 261 (1884). [Papers, i. 77.] 
{ A two-bladed screw-propeller of a ship is an example of a body of this kind. 



174 Motion of Solids through a Liquid [chap, vi 

The axis of x is one of the directions of permanent translation ; and is also 
the axis of one of the three screws of Art. 125, the pitch being —L/A. The 
axes of the two remaining screws intersect it at right angles, but not in 
general in the same point. 

7°. If, further, the body be identical with itself turned through one 
right angle about the above axis, the expression (6) must be unaltered when 
v, q, —w, —r are written for w, r, v, q, respectively. This requires that 
B=C,Q = R,P' = 0,M = N,F=0. Hence* 

2T = A u 2 + B (v 2 + w 2 ) +Pp 2 + Q(q 2 + r 2 ) + 2Lup + 2M (vq + wr). . . .(7) 

The form of this expression is unaltered when the axes of y, z are turned 
in their own plane through any angle. The body is therefore said to possess 
helicoidal symmetry about the axis of x. 

8°. If the body possess the same properties of skew symmetry about an 
axis intersecting the former one at right angles, we must evidently have 

2T=A(u 2 + v 2 + w 2 ) + P(p 2 + q 2 + r 2 )+2L(pu + qv + rw). ...(8) 
Any direction is now one of permanent translation, and any line drawn 
through the origin is the axis of a screw of the kind considered in Art. 125, 
of pitch —L/A. The form of (8) is unaltered by any change in the directions 
of the axes of co-ordinates. The solid is therefore in this case said to be 
' helicoidal ly isotropic' 

127. For the case of a solid of revolution, or of any other form to which 
the formula 

2T= Au 2 + B (v 2 + w 2 ) + Pp 2 + Q (q 2 + r 2 ) (1) 

applies, the complete integration of the equations of motion was effected by 
Kirchhofff in terms of elliptic functions. 

The particular case where the solid moves without rotation about its axis, 
and with this axis always in one plane, admits of very simple treatment!, and 
the results are very interesting. 

If the fixed plane in question be that of xy we have p, q, w = 0, so that the equations 
of motion, Art. 124 (1), reduce to 

A- T - = rBv, B-z- = —rAu, 

at dt ' 

, ! 12) 

Let x, y be the co-ordinates of the moving origin relative to fixed axes in the plane 
{xy) in which the axis of the solid moves, the axis of x coinciding with the line of the 

* This result admits of the same kind of generalization as (4), e.g. it applies to a body- 
shaped like a screw-propeller with three symmetrically-disposed blades. The integration of the 
equations of motion is discussed by Greenhill, "The Motion of a Solid in Infinite Liquid," 
Amer. Journ. of Math, xxviii. 71 (1906). 

t I.e. ante p. 160. 

% See Thomson and Tait, Art. 322; Greenhill, "On the Motion of a Cylinder through a 
Frictionless Liquid under no Forces," Mess, of Math. ix. 117 (1880). 



126-127] Solid of Revolution 1 75 

resultant impulse (/, say) of the motion ; and let 6 be the angle which the line Ox (fixed 
in the solid) makes with X. We have then 

Au = Icoa8, Bv= — isintf, r=6. 

The first two of equations (2) merely express the fixity of the direction of the impulse in 
space ; the third gives 

Q'6+ A ^I 2 sin6cos6 = (3) 

We may suppose, without loss of generality, that A>B. If we write 26 = $, (3) 
becomes 

^Tf*^ (4) 

which is the equation of motion of the common pendulum. Hence the angular motion of 
the body is that of a ' quadrantal pendulum,' i.e. a body whose motion follows the same 
law in regard to a quadrant as the ordinary pendulum does in regard to a half-circum- 
ference. When 8 has been determined from (3) and the initial conditions, X, y are to be 
found from the equations 

X = u cos 6 — v sin 8 = — cos 2 6 + -75 sin 2 8, 

.(5) 

(I I\ Q .. 

-j — -£ ) sin 8 cos 8= yd, 

the latter of which gives 

y=jt, (6) 

as is otherwise obvious, the additive constant being zero since the axis of X is taken to 
be coincident with, and not merely parallel to, the line of the impulse /. 

Let us first suppose that the body makes complete revolutions, in which case the first 
integral of (3) is of the form 

6 2 =<& 2 (l-k 2 sm 2 0), (7) 

where * 2= w3 (8) 

Hence, reckoning t from the position 0=0, we have 

*<<^Wi=w>' < 9 > 

in the usual notation of elliptic integrals. If we eliminate t between (5) and (7), and then 
integrate with respect to 6, we find 



{-L + t) f ^~t £ ^^ 



.(10) 



the origin of X being taken to correspond to the position 0=0. The path can then be 
traced, in any particular case, by means of Legendre's Tables. See the curve marked I on 
the next page. 

If, on the other hand, the solid does not make a complete revolution, but oscillates 
through an angle a on each side of the position 6 = 0, the proper form of the first integral 

of (3) is 

M x -S5' < n > 

, . „ ABO a> 2 

where sma== zri-72 (12) 



176 

If we put 

this gives 

whence 



Motion of Solids through a Liquid [chap, vi 



sin 6 = sin a sin yjr, 



^ 2= ^ « ( X ~ sin2 a sin2 +)> 
= i^(sino,Vr). 



sin 2 a 

cot 
sin a 




127-128] Solid of Revolution 111 

Transforming to yjr as independent variable, in (5), and integrating, we find 



.(14) 



.(17) 



X= -=- sin a . F (sin a, \ls) — -V cosec a . E (sin a, \i/-), 

£><£> 1 

y=-V cos^. 

The path of the point is now a sinuous curve crossing the line of the impulse at intervals 
of time equal to a half-period of the angular motion. This is illustrated by the curves III 
and IV of the figure. 

There remains a critical case between the two preceding, where the solid just makes a 
half-revolution, 6 having as asymptotic limits the two values ±^tt. This case may be 
obtained by putting k= 1 in (7), or a.=%ir in (1 1) ; and we find 

= cocos<9, (15) 

o>*=log tan (J «■ + ££), (16) 

x= 7r -logtan (i7r + £0)--^sin0, 

y=^cos0. 

See the curve II of the figure*. 

It is to be observed that the above investigation is not restricted to the case of a solid 
of revolution ; it applies equally well to a body with two perpendicular planes of sym- 
metry, moving parallel to one of these planes, provided the origin be properly chosen. If 
the plane in question be that of xy, then on transferring the origin to the point (FjB, 0, 0) 
the last term in the formula (2) of Art. 126 disappears, and the equations of motion take 
the form (2) above. On the other hand, if the motion be parallel to zx we must transfer 
the origin to the point ( - F/C, 0, 0). 

The results of this Article, with the accompanying diagram, serve to exemplify the 
statements made near the end of Art. 124. Thus the curve IV illustrates, with exaggerated 
amplitude, the case of a slightly disturbed stable steady motion parallel to an axis of per- 
manent translation. The case of a slightly disturbed unstable steady motion would be 
represented by a curve contiguous to II, on one side or the other, according to the nature 
of the disturbance. 

128. The mere question of the stability of the motion of a body parallel 
to an axis of symmetry may of course be treated more simply by approximate 
methods. Thus, in the case of a body with three planes of symmetry, as in 
Art. 126, 3°, slightly disturbed from a state of steady motion parallel to x, we 
find, writing u = u -f u, and assuming u', v, w, p, q, r to be all small, 
A du' _ „dv . „dw . \ 

A w= ' B dt = - Au « r ' °-dt^ Au ^ I (1) 

* In order to bring out the peculiar features of the motion, the curves have been drawn for 
the somewhat extreme case of A=5B. In the case of an infinitely thin disk, without inertia of 
its own, we should have A/B = co ; the curves would then have cusps where they meet the axis 
of y. It appears from (5) that x has always the same sign, so that loops cannot occur in any case. 

In the various cases figured the body is projected always with the same impulse, but with 
different degrees of rotation. In the curve I, the maximum angular velocity is J2 times what it 
is in the critical case II ; whilst the curves III and IV represent oscillations of amplitude 45° and 
18° respectively. 



178 Motion of Solids through a Liquid [chap, vi 

„ 6 e2V A(A-B) 2 . 
Hence ^~n&^ ~~R -tioV = 0, 

with a similar equation for r, and 

n d 2 w A(A-G) 2 . , ox 

a ^ + ^Q--V^ = 0, (2) 

with a similar equation for ^. The motion is therefore stable only when A 
is the greatest of the three quantities A, B, C. 

It is evident from ordinary Dynamics that the stability of a body moving parallel to an 
axis of symmetry will be increased, or its instability (as the case may be) will be diminished, 
by communicating to it a rotation about this axis. This question has been examined by 
Greenhill*. 

Thus, in the case of a solid of revolution slightly disturbed from a state of motion in 
which u and p are constant and the remaining velocities are zero, if we neglect squares 
and products of small quantities the first and fourth of equations (1) of Art. 124 give 

du/dt = 0, dp/dt**0, 

whence u = u , P=Po, (3) 

say, where w , p are constants. The remaining equations then take, on substitution from 
Art. 126 (3), the forms 



B \d~ P ° W ) = ~ A M ° r ' B \dt + p ° v ) = A U ° q ' 



•(4) 



Q^ t +(P-Q)por=-(A-B)uoW, Q ( t-(P-Q)p q = (A-B)u v (5) 

If we assume that v, w, q, r vary as e i<Tt , and eliminate their ratios, we find 

Q^±(P-2Q) Po cr-i [ (P-Q)p^ + ~(A-B)u ( ^=0 (6) 

The condition that the roots of this should be real is that 

PW + 4JjU-B)Qu i 

should be positive. This is always satisfied when A>B, and can be satisfied in any case 
by giving a sufficiently great value to p . 

This example illustrates the steadiness of flight which is given to an elongated projectile 
by rifling. 

129. In the investigation of Art. 125 the term 'steady' was used to 
characterize modes of motion in which the ' instantaneous screw ' preserved 
a constant relation to the moving solid. In the case of a solid of revolution, 
however, we may conveniently use the term in a somewhat wider sense, 
extending it to motions in which the vectors representing the velocities 
of translation and rotation are of constant magnitude, and make constant 
angles with the axis of symmetry and with each other, although their relation 
to points of the solid not on the axis may continually vary. 

* "Fluid Motion between Confocal Elliptic Cylinders, &c." Quart. Journ. Math. xvi. 227 (1879). 



128-130] Stability 179 

The conditions to be satisfied in this case are most easily obtained from the equations 
of motion of Art. 124, which become, on substitution from Art. 126 (4), 

dp 



.(1) 



A^B(rv-qw), P^ = 0, 

B^=Bpw-Aru, Q^=-(A-B)uw-(P-Q)pr, 

B^^Aqu-Bpv, Q^ = (A-B)uv + (P-Q)pq.^ 

It appears that p is in any case constant, and that q 2 + r 2 will also be constant provided 

v/q=wr, = &, say (2) 

This makes du/dt—0, and v 2 + w 2 = const. It follows that k will also be constant; and it 
only remains to satisfy the equations 

kB^{kBp-Au)r, Q ( k=-{( K A-B)ku + {P-Q)p}r. 

These will be consistent provided 

kB{(A-B)ku + (P-Q)p} + Q(kBp-Au) = 0, 

, u kBP /0 . 

whence -=T7i — 7 o P/ A — m ( 3 ) 

p AQ-k 2 B(A-B) 

Hence by variation of k we obtain an infinite number of possible modes of steady motion, 

of the kind above defined. In each of these the instantaneous axis of rotation and the 

direction of translation of the origin are in one plane with the axis of the solid. It is 

easily seen that the origin describes a helix about the line of the impulse. 

These results are due to KirchhofF. 

130. The only case of a body possessing helicoidal property, where simple 

results can be obtained, is that of the 'isotropic helicoid ' defined by Art. 126 (8). 

Let be the centre of the body, and let us take as axes of co-ordinates at any instant 
a line Ox parallel to the axis of the impulse, a line Oy drawn outwards from this axis, and 
a line Oz perpendicular to the plane of the two former. If I and K denote the force- and 
couple-constituents of the impulse, we have 

where w denotes the distance of from the axis of the impulse. 

Since AP—L 2 ^Q, the second and fifth of these equations shew that y = 0, ^=0. Hence 
■m is constant throughout the motion, and the remaining quantities are also constant ; in 
particular 



PI-LK LIm 



u = 



■(2) 



AP-L 2 ' AP-L 2 

The origin therefore describes a helix about the axis of the impulse, of pitch 

K P 

I L' 
This example is due to Kelvin*. 

* I.e. ante p. 168. It is there pointed out that a solid of the kind here in question may be 
constructed by attaching vanes to a sphere, at the middle points of twelve quadrantal arcs drawn 
so as to divide the surface into octants. The vanes are to be perpendicular to the surface, and 
are to be inclined at angles of 45° to the respective arcs. Larmor (I.e. ante p. 173) gives another 
example. "If... we take a regular tetrahedron (or other regular solid), and replace the edges 
by skew bevel faces placed in such wise that when looked at from any corner they all slope the 
same way, we have an example of an isotropic helicoid." 

For some further investigations in the present connection see a paper by Miss Fawcett, "On 
the Motion of Solids in a Liquid," Quart. Jcnirn. Math. xxvi. 231 (1893). 



180 Motion of Solids through a Liquid [chap, vi 

131. Before leaving this part of the Subject we remark that the preceding 
theory applies, with obvious modifications, to the acyclic motion of a liquid 
occupying a cavity in a mowng solid. If the origin be taken at the centre 
of inertia of the liquid, the formula for the kinetic energy of the fluid motion 
is of the type 

2T = m (u 2 + v 2 + w 2 ) + P^ 2 + Qq 2 + Rr 2 + 2V'qr + 2Q'rp + ZR'pq. . . .(1) 

For the kinetic energy is equal to that of the whole fluid mass (m), supposed 
concentrated at its centre of inertia and moving with this point, together with 
the kinetic energy of the motion relative to the centre of inertia. The latter 
part of the energy is easily proved by the method of Arts. 118, 121 to be 
a homogeneous quadratic function of p, q, r. 

Hence the fluid may be replaced by a solid of the same mass, having the 
same centre of inertia, provided the principal axes and moments of inertia be 
properly assigned. 

The values of the coefficients in (1), for the case of an ellipsoidal cavity, may be calcu- 
lated from Art. 110. Thus, if the axes of x, y, z coincide with the principal axes of the 
ellipsoid, we find 

P, Q> »-»«?=?. »«£#, *.£=#, P-, * R<=0. 

Case of a Perforated Solid. 

132. If the moving solid have one or more apertures or perforations, so 
that the space external to it is multiply-connected, the fluid may have 
a motion independent of that of the solid, viz. a cyclic motion in which the 
circulations in the several irreducible circuits which can be drawn through 
the apertures may have any given constant values. We will briefly indicate 
how the foregoing methods may be adapted to this case. 

Let k, k , k" , ... be the circulations in the various circuits, and let 8a, 8a', 
8a", ... be elements of the corresponding barriers, drawn as in Art. 48. 
Further, let I, m, n denote the direction-cosines of the normal, drawn towards 
the fluid at any point of the surface of the solid, or drawn on the positive 
side at any point of a barrier. The velocity-potential is then of the form 

</> + #o, 
where <j> = u<f> 1 + vfc + wfo + pxi + qx* + ^%3, \ ,-^ 

0o = KM -r tc co + k co + ... . J 

The functions cf> 1} cj> 2 , <$>s, %i, %2, %3 are determined by the same conditions as 
in Art. 118. To determine co, we have the conditions: (1°) that it must 
satisfy V 2 cw = at all points of the fluid ; (2°) that its derivatives must vanish 
at infinity; (3°) that dco/dn must = at the surface of the solid; and (4°) that 
co must be a cyclic function, diminishing by unity whenever the point to which 
it refers completes a circuit cutting the first barrier once (only) in the positive 



131-133] Perforated Solid 181 

direction, and recovering its original value whenever the point completes a 
circuit not cutting this barrier. It appears from Art. 52 that these conditions 
determine co save as to an additive constant. In like manner the remaining 
functions co , co", . . . are determined. 

By the formula (5) of Art. 55, twice the kinetic energy of the fluid is 
equal to 

-f»j] (♦ + *>£(♦ + *)*» 

-|«jJJ i <* + ««fa-^JU;<* + *)&»'- (2) 

Since the cyclic constants of <£ are zero, and since dfyojdn vanishes at the 
surface of the solid, we have, by Art. 54 (4), 

Hence (2) reduces to 

- P \\^ d S- pK \\^- pK -\\^-- (3) 

Substituting the values of </>, <£ from (1) we find that the kinetic energy 
of the fluid is equal to 

T + K, (4) 

where T is a homogeneous quadratic function of u, v, w, p, q, r, of the form 
defined by Art. 121 (2) (3), and 

2K=(tc,tc)K 2 + (K', k')k' 2 +... + 2(k,k , )kk' + ..., ...(5) 

where, for example, 

(*,*) = -p\\£d<r, 






> (6) 



The identity of the different forms of (k, k) follows from Art. 54 (4). 
Hence the total energy of fluid and solid is given by 

T = 1& + K, (7) 

where © is a homogeneous quadratic function of u, v, w, p, q, r of the same 
form as Art. 121 (8), and K is defined by (5) and (6) above. 

133. The 'impulse' of the motion now consists partly of impulsive forces 
applied to the solid, and partly of impulsive pressures pK, pK, pic", . . . applied 
uniformly (as explained in Art. 54) over the several membranes which are 
supposed for a moment to occupy the positions of the barriers. Let us 
denote by f 1} rji, £i, \ 1} fjL lf v\ the components of the extraneous impulse 



182 



Motion of Solids through a Liquid [chap, vi 



applied to the solid. Expressing that the ^-component of the momentum of 
the solid is equal to the similar component of the total impulse acting on it, 
we have 

= fi+ P {[("&+ ••• +PXi + ••• + Ka > + •••) -^dS 

-ft-S^U^f^J^^ w 

where, as before, T 2 denotes the kinetic energy of the solid, and T that part 
of the energy of the fluid which is independent of the cyclic motion. Again, 
considering the angular momentum of the solid about the axis of x, 

g- 1 = Xi - p J J (</> + fa) (ny - mz) 



dS 



= X 1 -™ + pK \\^ d S + pK '\\ a >' d £dS + (2) 



Hence, since ® = T + T 1} we have 

d® f f d<f> 



fi = 



du 



- pK \\ 



^dS-oK'Ha'^dS-.. 

on 1) on 



,d<fa 



.(B) 



By virtue of Lord Kelvin's extension of Greens Theorem, already referred 
to, these may be written in the alternative forms 



(4) 



Adding to these the terms due to the impulsive pressures applied to the 
barriers, we have, finally, for the components of the totil impulse of the 
motion *, 






X, fJL, V 



d® 



d®> 



d P +x °> ~^ + /X0) 



dr 



vo, 



.(5) 



where, for example, 

6 -^K(i + ^)^ + ^JJ(i + »b)^ + .... 

\o = ptc \\\7iy-mz+ -p\da + p/c' \\(ny - mz + ^M dor' + .... 

* Cf. Sir W. Thomson, I.e. ante p. 168 



L -(6) 



133-134] Components of Impulse 183 

It is evident that the constants f , rjo, £o> ^-o, Mo> v o are the components 
of the impulse of the cyclic fluid motion which would remain if the solid 
were, by forces applied to it alone, brought to rest. 

By the argument of Art. 119 ? the total impulse is subject to the same 
laws as the momentum of a finite dynamical system. Hence the equations 
of motion of the solid are obtained by substituting from (5) in the equations 
(1) of Art. 120* 

134. As a simple example we may take the case of an annular solid of 
revolution. 

If the axis of x coincide with that of the ring, we see by reasoning of the same kind as 
in Art. 126, 4° that if the situation of the origin on this axis be properly chosen we may 
write 

2T=Au 2 + B(v 2 + iv*) + Pp 2 + Q(q 2 + r 2 ) + ( K , <) k 2 (1) 

Hence g, rj. £=Au + $ , Bv, Bw; X, /*, v = Pp, Qq, Qr (2) 

Substituting in the equations of Art. 120, we find dp/dt = 0, or p = const., as is other- 
wise obvious. Let us suppose that the ring is slightly disturbed from a state of motion in 
which v, w, p, q, r are zero, i.e. a steady motion parallel to the axis. In the beginning of 
the disturbed motion v, w, p, q, r will be small quantities whose products we may neglect. 
The first of the equations referred to then gives du/dt — 0, or u= const., and the remaining 
equations become 



.(3) 



B d J r -(Au + ^)r, Q d J-=- {{A -B)u+£ Q }w, 

B d f t = (Au + £ )q, Q<^ = {(A-B)u + £ }v. 

Eliminating r, we find 

BQ C ^==-(Au + Z ){(A-B)u + £ }v (4) 

Exactly the same equation is satisfied by w. It is therefore necessary and sufficient for 
stability that the coefficient of v on the right-hand side of (4) should be negative ; and the 
time of a small oscillation, when this condition is satisfied, ist 

2 r BQ -u () 

L(Au + £ ){(A-B)u + &}_\ {) 

We may also notice another case of steady motion of the ring, viz. where the impulse 
reduces to a couple about a diameter. It is easily seen that the equations of motion are 
satisfied by £, rj, £, X, /x = 0, and v constant; in which case 

u — — £JA , r = const. 

The ring then rotates about an axis in the plane yz parallel to that of 2, at a distance ujr 
from it J. 

* This conclusion may be verified by direct calculation from the pressure-formula of Art. 20 ; 

see Bryan, " Hydrodynamical Proof of the Equations of Motion of a Perforated Solid, ," 

Phil. Mag. (5) xxxv. 338 (1893). 

t Sir W. Thomson, I.e. ante p. 168. 

% For further investigations on this subject we refer to papers by Basset, "On the Motion 
of a King in an Infinite Liquid," Proc. Camb. Phil. Soc. vi. 47 (1887), and Miss Fawcett, I.e. ante 
p. 179. 



184 Motion of Solids through a Liquid [chap, vi 

The Forces on a Cylinder moving in Two Dimensions. 

134 a. The two-dimensional problem of the motion of a cylindrical body, 
especially when there is circulation round it, is most simply treated by direct 
calculation of the pressures on the surface*. We assume as usual that the 
fluid is at rest at infinity. 

Taking axes fixed in a cross-section, we denote by (u, v) the velocity of 
the origin, and by r the angular velocity, the symbols u, v being now required 
in their original sense as component velocities of the fluid. The pressure- 
equation is then 

;-g-(«-nr)g-(T + »)|-w + «™t (i) 

where q 2 = u 2 + v 2 . The force (X, Y) and couple (N) to which the pressures 
on the surface reduce are 

X = —\plds, Y——\pmds, N = — \p (mx — ly) ds, (2) 

where I, m are the direction cosines of the normal drawn outwards from an 
element hs of the contour, and the integration is taken round the perimeter. 
Now 

^ \q 2 lds = — ( u £— + v jr- J dxdy = \(lu + mv) uds, 

-£ q 2 mds = — M tt — + ^ — J dxdy = \(lu + mv) vds 

in virtue of the relations 

dv/dx as du/dy, du/dx + dv/dy — 0. 
We have here omitted the various line-integrals taken over an infinite enclosing 
boundary, since at a great distance r the velocity is at most of the order 1/r, 
whilst 8s is of the order rB0. At the surface of the cylinder we have 

lu + mv — I (u — ry) + m (v + rx) (4) 

Hence substituting from (1) in (2) we find 
X 
P 



(3) 



= — \~ Ids + (mu — Iv) (v + rx) ds 

— J|*M. + ](* + ») f*A (5) 

and similarly 

= -]^mds-](Ti-ry)^ds (6) 



P 

Again we find, 



J q 2 (mx — ly) ds = \(lu + mv) (xv — yu) ds (7) 

* Aeronautical Research Committee, R. and M. 1218 (1929). For another treatment see Glauert, 
R. and M. 1215 (1929). 



134 a] 



Forces on a Moving Cylinder 



185 



Here also, the line-integrals round an infinitely remote boundary are omitted, 
since we may suppose that at this boundary l/x = m/y, and that lu + mv is of 
the order 1/r 2 . The formula (2) for N thus becomes 

— = — \~ (moe — ly) ds + (ux + vy) (Iv — mu) ds 

= -^(mx-ly)ds-j(jix + vy)^ds (8) 

We now write, in analogy wifch Arts. 118, 132, 

<£ = !!(/>! + v0 2 + r;\; + </>o, (9) 

where (j> represents the circulatory motion which would persist if the cylinder 
were brought to rest. It is therefore a cyclic function with, say, the cyclic 
constant k. Comparing with (4) we have, at the surface of the cylinder, 



dn 



h 



d</>2 

dn 



m> d ^=-{mx-ly\ d -p=0 (10) 

dn J dn 7 



In the absence of circulation the energy of the fluid would be 

T = -ipj(<t>-<f>o) d ^ds (11) 

Substituting from (9) and (10) this gives 

2T = Au 2 + 2Huv + Bv 2 + Rr 2 + 2 (Lu 4- Mv) r, (12) 



where 



A=p \lfads, H = p \lfads = p \mfads, "B — p \mfads 
p]{mx-ly) x ds, 
p \lxds — p (mx — ly) <$> x ds, M = p\ m^ds = p (mx — ly) <f> 2 ds. 



The leading terms in (5), (6), and (8) now take the forms 

--(Au + Hv + Lr) = -^, 

d /TT „ __ . d dT 

-^(Hu + Bv + Mr) — g^, 



.(13) 



-^(Rr + Lu+Mv) 



ddT 

dtdr' 



.(14) 



Again, we have 

p \x ^~ E^ ds = p L ((/)-</) )^ = Hu + Bv + Mr=~ 



p\y 



a(0-^ >o) 

ds 



ds 



dT 

^(</>-</)o)^ = -(Au + HvH-Lr) = -^ i . 



(15) 



186 Motion of Solids through a Liquid 

Hence if we write 

d<f>o 



[CHAP. VI 



J-?*-* y-t ds =e> < 16 > 



the expressions for the forces become 



X = 



Y = - 



ddT dT 

dt du dv 



Kpv + par, 



_ddr 

dtdv 



dT 
r^ + icpu + pPr, 

dT 



dT 



•(17) 



„ ddT 

N = -^8r- + V 8H- U 8T-^ aU + ^ v) \ 
By turning the co-ordinate axes through a suitable angle, the coefficient 
H can be made to vanish. And by a suitable choice of origin we may also 
annul the coefficients L, M, or alternatively we may make a = 0, j3 = 0. But 
these two determinations are in general incompatible, and neither of these 
special origins can be assumed to coincide with the mean centre of the area 
of the section. 

The most interesting case, however, is where the section is symmetrical 
with respect to each of two perpendicular axes. If these are taken as axes of 
co-ordinates we have 

H=0, L = 0, M = 0, a = 0, = 0, (18) 

and the formulae (17) reduce to 



X 



- A -J- + Brv 
dt 

dv 



tcv, 



.(19) 



Y = — B t- — Aru + ku, 

N = -R|-(A-B)uv. 

To form the equation of motion in this case we have only to modify the 
inertia coefficients, as in Art. 122. If the distribution of mass is also 
symmetrical, we write 

A = A + M, B = B + M, R = R + L, (20) 

where M represents the mass of the cylinder itself, and L its moment of 
inertia. Then 

fin 



a du „ 
A Tt -Brv + «pv 



.(21) 



B -T- + ^Iru — /cpu = Y, 

dr 
L~-(A-B)xw=N, 

where X y Y, N represent the effect of extraneous forces. When these are 
absent, and the circulation zero, the solution is as in Art. 127. 



i34a-i35] Generalized Co-ordinates 187 

In the case of a circular section there is no point in supposing the 
co-ordinate axes to rotate. Putting A — B, r = 0, we have 

A d f t+Kp u = X, A^-cpn-Y, (22) 

as in Art. 69. 

If the section is symmetrical with respect to one axis only, say that of x, 
we have H = 0, L = 0, B — 0. By a displacement of the origin along the axis 
of symmetry we can make M = 0,but a will not in general vanish simultaneously. 
If there is no circulation the new origin corresponds to the c centre of reaction ' 
of Thomson and Tait*. 

Equations of Motion in Generalized Co-ordinates. 

135. When we have more than one moving solid, or when the fluid is 
bounded, wholly or in part, by fixed walls, we may have recourse to Lagrange's 
method of ' generalized co-ordinates.' This was first applied to hydrodynamical 
problems by Thomson and Tait j*. 

The systems ordinarily contemplated in Analytical Dynamics are of finite 
freedom ; i.e. the position of every particle is completely determined when 
we know the values of a finite number of independent variables or ' generalized 
co-ordinates' q lt q 2 , ... q n . The kinetic energy T can then be expressed as a 
quadratic function of the 'generalized velocity components' q lt q 2 , ••• q n . 

In the Hamiltonian method the actual motion of the system between any 
two instants t , t-i is compared with a slightly varied motion. If f, rj, f be the 
Cartesian co-ordinates of any particle m, and X, Y, Z the components of the 
total force acting on it, it is proved that 

P l {AT+2(XAf+ YArj + ZAtydt^O, (1) 

provided the varied motion be such that 



2m(%AZ + riAy + JA?) 



= (2) 



The summation 2 is understood to include all the particles of the system. 
The varied motion is usually supposed to be adjusted so that the initial 
and final positions of each particle shall be respectively the same as in the 
actual motion. The quantities Af, At;, Af then vanish at each limit of 
integration, and the condition (2) is fulfilled. 

For a conservative system free from extraneous force (1) takes the form 



A \ t \T-V)dt = (3) 



* Natural Philosophy, Art. 321. 
t Ibid. Art. 331, 



188 Motion of Solids through a Liquid [chap, vi 

In words, if the actual motion of the system between any two configura- 
tions through which it passes be compared with any slightly varied motion, 
between the same configurations, which the system is (by the application of 
suitable forces) made to execute in the same time, the time-integral of the 
'kinetic potential '* V— T is, stationary. 

In terms of generalized co-ordinates, the equation (1) takes the form 

\ t \&T+Q 1 Aq 1 + Q 2 &q 2 +... + Q n &q n )dt = 0, (4) 

from which Lagrange's equations 

d dT dT _ ~ ,p,. 

dtdir" d^ r ~ Vr W 

can be deduced by a known process. 

136. Proceeding now to the hydrodynamical problem, let q lt q 2 , ... q n be 
a system of generalized co-ordinates which serve to specify the configuration 
of the solids. We will suppose, for the present, that the motion of the fluid 
is entirely due to that of the solids, and is therefore irrotational and acyclic. 

In this case the velocity-potential at any instant will be of the form 

<£ = 9i0i + ?202+ ... +q n <t>n> (1) 

where </>i, <j> 2 , ... are determined in a manner analogous to that of Art. 118. 
The formula for the kinetic energy of the fluid is then 



2T = - /J Jj</,^^=A 11 gi 2 + A 22 g 2 2 +...+2A 12 ^ 2 +..., (2) 

Kr=~p J J* r ^ dS, A rs = -p j|</> r g dS = -p J J*. ^ dS, . . .(3) 



where 



the integrations extending over the instantaneous positions of the bounding 
surfaces of the fluid. The identity of the two forms of A rs follows from 
Green's Theorem. The coefficients A rr , A rs will in general be functions of 
the co-ordinates q 1} q 2 , ... q n . 

If we add to (2) twice the kinetic energy, Ti, of the solids themselves, we 
get an expression of the same form, with altered coefficients, say 

2T = Autf + A 22 q 2 * + ... +2A u q 1 q 2 + (4) 

It remains to shew that, although our system is one of infinite freedom, 
the equations of motion of the solids can, under the circumstances pre- 
supposed, be obtained by substituting this value of T in the Lagrangian 
equations, Art. 135 (5). We are not at liberty to assume this without 
further examination, for the positions of the various particles of the fluid are 

* The ndime was introduced by Helmholtz, "Die physikalische Bedeutung des Princips der 
kleinsten Wirkung," Crelle, c. 137, 213 (1886) [Wiss. Abh. iii. 203]. 



135-136] Application to Hydrodynamics 189 

not determined by the instantaneous values q lf q 2 , ... q n of the co-ordinates 
of the solids. For instance, if the solids, after performing various evolutions, 
return each to its original position, the individual particles of the fluid will 
in general be found to be finitely displaced*. 

Going back to the general formula (1) of Art. 135, let us suppose that in 
the varied motion, to which the symbol A refers, the solids undergo no 
change of size or shape, and that the fluid remains incompressible, and has, 
at the boundaries, the same displacement in the direction of the normal as 
the solids with which it is in contact. It is known that under these 
conditions the terms due to the internal reactions of the solids will disappear 
from the sum 

The terms due to the mutual pressures of the fluid elements are equivalent to 

or lip (IA% + mA v + nA£) dS + [lip (^ + ^ + d -^\ dxdydz, 

where the former integral extends over the bounding surfaces, and I, m, n 
denote the direction-cosines of the normal, drawn towards the fluid. The 
volume-integral vanishes by the condition of incompressibility 

¥ f + T ? + X ? -0 < 5 > 

ox ay dz 

The surface-integral vanishes at a fixed boundary, where 

ZAf + mAr) + nA£ = 0; 

and in the case of a moving solid it is cancelled by the terms due to the 
pressure exerted by the fluid on the solid. Hence the symbols X, Y, Z may 
be taken to refer only to the remaining forces acting on the system, and we 
may write 

Z(XA£+YA v + ZAO=QiA qi + Q 2 Aq 2 + ... + Q n Aq n , (6) 

where Q 1} Q 2} ... Q n are generalized components of force. 

The varied motion of the fluid has sfcill a high degree of generality. We 
will now further limit it by supposing that while the solids are, by suitable 
forces applied to them, made to execute an arbitrary motion, the fluid is left 
to take its own course in consequence of this. The varied motion of the 
fluid may accordingly be taken to be irrotational, in which case the varied 
kinetic energy T+AT of the system will be the same function of the 
varied co-ordinates q r + Aq r , and the varied velocities q r + Aq r , that the 
actual energy T is of q r and q r . 

* As a simple example, take the case of a circular disk which is made to move, without 
rotation, so that its centre describes a rectangle two of whose sides are normal to its plane ; and 
examine the displacements of a particle initially in contact with the disk at its centre. 



.7)1 



190 Motion of Solids through a Liquid [chap, vi 

Again, considering the particles of the fluid alone, we shall have, on the 
same supposition, 

2m (£Af + v^v + SAf) = - P jjj(g A* + g Ay + g A*) fofy* 

= p [(<£ (ZAf + rnA?? + wAf) <Z£, 

where use has again been made of the condition (5) of incompressibility. By 
the kinematical condition to be satisfied at the boundaries, we have 

and therefore 

= (Aii^ + A 12 q 2 + • • • + A ln q n ) Aq x + (Aaji + A2252 + . ■ • + A 2?l ^n) Ag 2 

-f ... +(A n i^ 1 + A n2 J2+ ... +A nn ^ n )A^ n 

-s* Aft t5i^ + - + "8S;^ (7) 

by (1), (2), (3) above. If we add the terms due to the solids, we find that 
the condition (2) of Art. 135 still holds; and the deduction of Lagrange's 
equations 

then proceeds in the usual manner. 

137. As a first application of the foregoing theory we may take an 
example given by Thomson and Tait*, where a sphere is supposed to move 
in a liquid which is limited only by an infinite plane wall. 

Taking, for simplicity, the case where the centre moves in a plane perpendicular to 
that of the wall, let us specify its position at time t by rectangular co-ordinates x, y in this 
plane, of which y denotes distance from the wall. We have 

2T=Ax 2 + By 2 , (1) 

where A and B are functions of y only, it being plain that the term xy cannot occur, since 
the energy must remain unaltered when the sign of x is reversed. The values of A , B can 
be written down from the results of Arts. 98, 99, viz. if m denote the mass of the sphere, 
and a its radius, we have 

J^«+|,rpa*(l + AJjJ), *«m+i«r^(r + t|£), (2) 

approximately, if y be great in comparison with a. 
The equations of motion give 

>>=* >>-i(f* 2+ S^H (3) 

where X, Y are the components of extraneous force, supposed to act on the sphere in a 
line through the centre. 

* Natural Philosophy , Art. 321. 



136-138] Application to Hydrodynamics 191 

If there be no extraneous force, and if the sphere be projected in a direction normal to 
the wall, we have x=0, and 

By 2 =const (4) 

Since B diminishes as y increases, the sphere experiences an acceleration from the wall. 

Again, if the sphere be constrained to move in a line parallel to the wall, we have 
y = 0, and the necessary constraining force is 

y =-if* 2 - < 5 > 

Since dAjdy is negative, the sphere appears to be attracted by the wall. The reason of 
this is easily seen by reducing the problem to one of steady motion. The fluid velocity 
will evidently be greater, and the pressure therefore less, on the side of the sphere next 
the wall than on the further side ; see Art. 23. 

The above investigation will also apply to the case of two spheres projected in an 
unlimited mass of fluid, in such a way that the plane y==0 is a plane of symmetry in all 
respects. 

138. Let us next take the case of two spheres moving in the line 

of centres. 

The kinematical part of this problem has been treated in Art. 98. If we now denote 
by x, y the distances of the centres of the spheres A, B from some fixed origin in the 
line joining them, we have 

2T=Lx 2 -2Mxy + JYy\ (1) 

where the coefficients Z, M t N are functions of y — #, or c, the distance between the centres. 
Hence the equations of motion are 

d . ,,. ,,.. . (dL .„ a dM . . dN .A „ 

where X, Y are the forces acting on the spheres along the line of centres. If the radii a, b 
are both small compared with c, we have, by Art. 98 (15), keeping only the most important 
terms, 

L=m + lTrpd\ M^Zirp — , iV=m' + f 7rp& 3 , (3) 

approximately, where m, m' are the masses of the two spheres. Hence to this order of 
approximation 

dL n dM a?b* dN „. 

^ = ' ^ = - 6 ^^-' -aTc =0 - 

If each sphere be constrained to move with constant velocity, the force which must be 
applied to A to maintain its motion is 

This tends towards B, and depends only on the velocity of B. The spheres therefore 
appear to repel one another ; and it is to be noticed that the apparent forces are not equal 
and opposite unless x=±y. 

Again, if each sphere make small periodic oscillations about a mean position, the period 
being the same for each, the mean values of the first terms in (2) will be zero, and the 
spheres therefore will appear to act on one another with forces equal to 

top^g-m -(5) 



■(2) 



192 Motion of Solids through a Liquid [chap, vi 

where [xy] denotes the mean value of xy. If x, y differ in phase by less than a quarter- 
period, this force is one of repulsion, if by more than a quarter-period it is one of attraction. 

Next, let B perform small periodic oscillations, while A is held at rest. The mean force 
which must be applied to A to prevent it from moving is 

X=ilw («) 

where [y 2 ] denotes the mean square of the velocity of B. To the above order of approxi- 
mation dN/dc is zero ; on reference to Art. 98 we find that the most important term in it 
is — 127rpa 3 6 6 /c 7 , so that the force exerted on A is attractive, and equal to 

^p-fm (7) 

This result comes under a general principle enunciated by Kelvin. If we have two 
bodies immersed in a fluid, one of which {A ) performs small vibrations while the other (B) 
is held at rest, the fluid velocity at the surface of B will on the whole be greater on the 
side nearer A than on that which is more remote. Hence the average pressure on the 
former side will be less than that on the latter, so that B will experience on the whole an 
attraction towards A. As practical illustrations of this principle we may cite the apparent 
attraction of a delicately-suspended card by a vibrating tuning-fork, and other similar 
phenomena studied experimentally by Guthrie* and explained in the above manner by 
Kelvin t. 

Modification of Lagrange's Equations in the case of Cyclic Motion. 

139. We return to the investigation of Art. 136, with the view of 
adapting it to the case where the fluid has cyclic irrotational motion through 
channels in the moving solids, or (it may be) in an enclosing vessel, in- 
dependently of the motion due to the solids themselves. 

Let us imagine barrier-surfaces to be drawn across the several apertures. 
In the case of channels in a containing vessel we shall suppose these ideal 
surfaces to be fixed in space, and in the case of channels in a moving solid 
we shall suppose them to be fixed relatively to the solid. Let ^, #', % ', ... 
be the fluxes at time t across, and relative to, the several barriers ; and let 
X> X> X> •" De * ne time-integrals of these fluxes, reckoned from some 
arbitrary epoch, these quantities determining (therefore) the volumes of 
fluid which have up to the time t crossed the respective barriers. It will 
appear that the analogy with a dynamical system of finite freedom is still 
conserved, provided the quantities %, %', %", ... be reckoned as generalized 
co-ordinates of the system, in addition to those (q x , q 2 , ... q n ) which specify 
the positions of the moving solids. It is obvious already that the absolute 
values of x> X> X> ••• w ^ not en t er into the expression for the kinetic 
energy, but only their rates of variation. 

In the first place, we may shew thai; the motion of the fluid, in any given 
configuration of the solids, is completely determined by the instantaneous 

* "On Approach caused by Vibration," Phil. Mag. (4) xl. 345 (1870). 

t Reprint of Papers on Electrostatics, dbc. Art. 741. For references to further investigations, 
both experimental and theoretical, by C. A. Bjerknes and others, on the mutual influence of 
oscillating spheres in a fluid, see Hicks, "Eeport on Kecent Kesearcb.es in Hydrodynamics," Brit. 
Ass. Rep. 1882, pp. 52...; Love, Encycl. d. math. Wiss. iv. (3), pp. Ill, 112. 



138-139] Hamiltonian Method 193 

values of q lf q 2 , • •• q n > % X '» X> -^ or ^ there were two modes of 

irrotational motion consistent with these values, then, in the motion which 
is the difference of these, the boundaries of the fluid would be at rest, and 
the flux across each barrier would be zero. The formula (5) of Art. 55 shews 
that under these conditions the kinetic energy would vanish. 

It follows that the velocity-potential can be expressed in the form 

</> - $i*i + q*<t>2+ ... +4n<l>n + xto+x' n '+ W 

Here cb r is the velocity-potential of a motion in which q r alone varies and 
the flux across each barrier is accordingly zero. Again O is the velocity- 
potential of a motion in which the solids are all at rest, whilst the flux 
through the first aperture is unity, and that through every other aperture is 
zero. It is to be observed that <f> 1} <£ 2 , ••• </> n > &, &', ••• are in general all of 
them cyclic functions, which may however be treated as single-valued, on the 
conventions of Art. 50. 

The kinetic energy of the fluid is given by the expression 



2T 



'///{©'♦S)'*®}*** < 2 > 



where the integral is taken over the region occupied by the fluid at the 
instant under consideration. If we substitute from (1) we obtain T as a 
homogeneous quadratic function of q 1} q 2 , ... q n , %, %', x > ••• w ^ n coefficients 
which depend on the instantaneous configuration of the solids, and are there- 
fore functions of q 1} q 2 , ... q n only. Moreover, we find, by Art. 53 (1), 

where k, k, ... are the cyclic constants of (/>, and the first surface -integral is 
to be taken over the surfaces of the solids, and the remaining ones over the 
several barriers. By the conditions which determine 12, this reduces to the 
first equation of the system : 

dT dT , 

%-'*' ty= pK > (3) 

These shew that p/c, pre', . . . are to be regarded as the generalized components 
of momentum corresponding to the velocity-components %, ^', ..., respec- 
tively. 

We have recourse to the general Hamiltonian formula (1) of Art. 135. 
We will suppose that the varied motion of the solids is subject only to the 
condition that the initial and final configurations are to be the same as in the 
actual motion ; also that the initial position of each particle of the fluid is the 
same in the two motions. The expression 

2m(fAf + i/Ai7 + £A£) ..(4) 



194 



Motion of Solids through a Liquid [chap, vi 



will accordingly vanish at time t , but not in general at time t ly in the absence 
of further restrictions. 

We will now suppose that the varied motion of the fluid is irrotational, and 
accordingly determined by the instantaneous values of the varied generalized 
co-ordinates and velocities. Considering the particles of the fluid alone, we 
have 

= p \\<l>(lAg+mAii+nA£)dS+pK [|(ZAf + mAr) + nA£) do- 

+ P k' [l(lA£ + mA v +nAZ)d<r' + ..., (5) 

where I, m, n are the direction-cosines of the normal to an element of the 
bouuding surface, drawn towards the fluid, or (as the case may be) of the 
normal to an element of a barrier, drawn in the direction in which the 
corresponding circulation is estimated. 

At time t± we shall have 

ZAf + mA77 + ?iAf=0 

at the surface of the solids, as well as at the fixed boundaries. Again, if A B 
represent one of the barriers in its position at time t ly 
and if A f B' represent the locus at the same instant, in 
the varied motion, of those particles which in the actual 
motion occupy the position AB, the volume included 
between AB and A'B' will be equal to the corre- 
sponding A%, whence 




.(6) 



(ZA£ + mA?7 4- nA? ) da = A%, 
[ [ (ZAf -I- m&y + **Af ) da' = A%', 



The varied circulations are, from instant to instant, still at our disposal. 
We may suppose them to be so adjusted as to make A%, A^', ... vanish at 
time ti. The expression (4) will accordingly vanish, and if we further suppose 
that the external forces do on the whole no work when the boundary of the 
fluid is at rest, whatever relative displacements be given to the parts of the 
fluid, we have 

f' , {AT+Q 1 A ?1 + Q 2 Ag 2 + ...+Q n A^)^ = 0, (7) 

s before. 

By a partial integration, and remembering that by hypothesis 
Aq ly Aq 2 , ... A^, A % , A*', ... 



139-140] Ignoration of Co-ordinates 195 

vanish at the limits t ,t ly but are otherwise independent, we obtain n equations 
of the type 

dtdq r dq r ~^ W 

togetherwith ||f = > S |f " °« (9) 

140. Equations of the type (8) and (9) present themselves in various 
problems of ordinary Dynamics, e.g. in questions relating to gyrostats, where 
the co-ordinates %, %', . .., whose absolute values do not affect the kinetic or the 
potential energy of the system, are the angular co-ordinates of the gyrostats 
relative to their frames. The general theory of such systems has been treated 
by Routh*, Thomson and Taitf, and other writers. 

We have seen that -^-.—pK, ~-r, = pi<! y . . . , (10) 

and the integration of (9) shews that the quantities k, k, ... are constants 
with regard to the time, as is otherwise known (Art. 50). Let us write 

R = T-p K x-p«'x- (11) 

The equations (10), when written in full, determine %, %', ... as linear functions 
of k, k, ... and q ly q 2y ... q n \ and by substitution in (11) we can express R as 
a homogeneous quadratic function of the same quantities, with coefficients 
which of course in general involve the co-ordinates q ly q 2y ... q n . On this 
supposition we have, performing an arbitrary variation & on both sides of (11), 
and omitting terms which cancel by (10), 

dR ^ dR * ' dR 5, 

=-r dgi + . . . + ^— dtfi + . . . + ^— OK + . . . 

= ||Sg 1 + ... + ||85 1+ ...-^8*- (12) 

where, for brevity, only one term of each kind is exhibited. Hence we obtain 
2n equations of the types 

dR = dT dR^dT 

dq r dq r ' dq r dq r ' 

togetherwith _ = - p ^, — , = -px', (14) 

Hence the equations (8) may be written 

d dR dR _ ^ ,,.v 

dt dq r dq r 

* On the Stability of a Given State of Motion (Adams Prize Essay), London, 1877; Advanced 
Rigid Dynamics, 6th ed., London, 1905. 

f Natural Philosophy, 2nd ed., Art. 319 (1879). See also Helmholtz, "Principien der Statik 
monocyclischer Systeme," Crelle, xcvii. (1884) [Wiss. Abh. iii. 179]; Larmor, "On the Direct 
Application of the Principle of Least Action to the Dynamics of Solid and Fluid Systems," Proc. 
Lond. Math. Soc. (1) xv. (1884) [Papers, i. 31]; Basset, Proc. Camb. Phil. Soc. vi. 117 (1889). 



196 



Motion of Solids through a Liquid [chap, vi 



where the velocities %, %', ... corresponding to the 'ignored' co-ordinates 
%, %', ... have now been eliminated*. 

141. In order to shew more explicitly the nature of the modification 
introduced by the cyclic motions into the dynamical equations, we proceed as 
follows. 

If we substitute in (11) from (14), we obtain 

y =*-("g + *'g + -) < 16 > 

Now, remembering the composition of R, we may write for a moment 

R — ^2,0 + R\,l + ^0,2 j (17) 

where R 2y o is a homogeneous quadratic function of q x , q%, ... q n , i2 0)2 is a 
homogeneous quadratic function of k, k\ ... , and R lf i is bilinear in these two 
sets of variables. Hence (16) takes the form 

T — i?2,0 — ^0,2 > (18) 

or, as we shall henceforth write it, 

t=® + k, (19) 

where tlT and K are homogeneous quadratic functions of q lt q%, ... q n , and of 
k, k, ... , respectively. It follows also from (17) that 

r = i&-r- fab- fab- ...- p«in, (20) 

where /3i , /?2 > • • • are linear functions of k, k, ... , say 

/3 1 = a 1 K + a 1 'tc' + ... , * 



/3„ = a w /c + a w V + 



The meaning of the coefficients a (in the hydrodynamical problem) appears 
from (14) and (20). We find 



.(22) 



. dK . 
PX = d/c + a i?i + «2<?2 + ... 4- a n ?r 

., dK ,. 

px = ^7 + «igi + «2?2+... +« w g r » 



which shew that a r is the contribution to the flux of matter across the first 
barrier due to unit rate of variation of the co-ordinate q ry and so on. 

If we now substitute from (20) in the equations (15) we obtain the general 
equations of motion of a ' gyrostatic system,' in the formf 

* This investigation is due to Bouth, I.e.; cf. Whittaker, Analytical Dynamics, Art. 38. 

t These equations were first given in a paper by Sir W. Thomson, "On the Motion of Kigid 
Solids in a Liquid circulating irrotationally through perforations in them or in a Fixed Solid," 
Phil. Mag. (4) xlv. 332 (1873) [Papers, iv. 101]. See also C. Neumann, Hydrodynamische 
Untersuchungen (1883). 



ho-142] Kineto- Statics 197 

lf-| + o.«)* + a.8)* + .. r+ a»)*. + |-ft. 

(23) 

where C'.')-fr-t <"> 

It is important to notice that (r, s) = — (5, r), and (r, r) = 0. 

If in the equations of motion of a fully-specified system of finite freedom 
(Art. 135 (4)) we reverse the sign of the time-element Bt, the equations are 
unaltered. The motion is therefore reversible ; that is to say, if as the system 
is passing through any assigned configuration the velocities qi , q 2 , • . • q n be all 
reversed, it will (if the forces be always the same in the same configuration) 
retrace its former path. It is important to observe that this statement does 
not in general hold of a gyros tatic system ; thus, the terms in (23) which are 
linear in q lt q 2 , ... q n change sign with 8t, whilst the others do not. Hence, in 
the present application, the motion of the solids is not reversible, unless indeed 
we imagine the circulations tc, k, ... to be reversed simultaneously with the 
velocities q lf q 2 , ... q n *• 

If we multiply the equations (23) by q lt q 2 , ... q n in order, and add, we find, 
by an easy adaptation of the usual process, 

g(« + JO = &$! + && +... + &$«, (25) 

or, if the system be conservative, 

*& + !{+¥= const (26) 

142. The results of Art. 141 may be applied to find the conditions of 
equilibrium of a system of solids surrounded by a liquid in cyclic motion. 
This problem of ' Kineto-Statics/ as it may be termed, is however more 
naturally treated by a simpler process. 

The value of <f> under the present circumstances can be expressed in the 
alternative forms 

#-xP+#d'.+ ..., (i) 

(j) = fCCO + K(o' + . . . ; . (2) 

and the kinetic energy can accordingly be obtained as a homogeneous quad- 
ratic function either of %%> ... , or of k, k, ... , with coefficients which are 
in each case functions of the co-ordinates q 1} q 2 , ... q n which specify the 

* Just as the motion of the axis of a top cannot be reversed unless we reverse the spin. 



198 Motion of Solids through a Liquid [chap, vi 

configuration of the solids. These two expressions for the energy may be 
distinguished by the symbols T and K, respectively. Again, by Art. 55 (5) 
we have a third formula 

2T= P kx + PI c'x' + (3) 

The investigation at the beginning of Art. 139, shortened by the omission 
of the terms involving q ly q 2 , ... q n , shews that 

dT , dT 
P«=g£. P"=W' (4) 

Again, the explicit formula for K is 

= (*, K)K 2 + (fc', fc')K' 2 +...+2(/C ) K')fCK'+..., (5) 



where 



= (tc, k) k + (/c, k) k + . . . = - p Lp do: 



and so on. Hence 

dK 

die 

We thus obtain p % = — ) p tf = — , (7) 

Again, writing T + K for IT in (3), and performing a total variation 8 on 
both sides of the resulting identity, we find, on omitting terms which cancel 
in virtue of (4) and (7)*, 

^ + ^=0 (8) 

dq r dq r 

This completes the requisite analytical formulae f. 

If we now imagine the solids to be guided from rest in the configuration 
(<2i, <}2> ••• <?n) to rest in an adjacent configuration 

(#i + Aq 1} q 2 + Aq 2 , ... q n + Aq n ), 

the work required is Qi&qi + Qi^q 2 4- ... + Q n &q n , 

where Q iy Q 2 , •• Q n are the components of extraneous force which have to be 
applied to neutralize the pressures of the fluid on the solids. This must be 
equal to the increment AK of the kinetic energy, calculated on the supposition 
that the circulations k, tc', ... are constant. Hence 

<2'=af W 

* It would be sufficient to assume either (4) or (7) ; the process then leads to an independent 
proof of the other set of formulae. 

t It may be noted that the function R of Art. 140 now reduces to - K. 



142-143] Kineto-Statics 199 

The forces representing the pressures of the fluid on the solids (when these 
are held at rest) are obtained by reversing the signs, viz. they are given by 

*— .£; (10) 

the solids therefore tend to move so that the kinetic energy of the cyclic 
motion diminishes. 

In virtue of (8) we have, also, 

Q/ = ^° • (11) 

dq r 

143. The formula (19) of Art. 141 may be applied to find approximate 
expressions for the forces on a solid immersed in a non-uniform stream*. 

Suppose we have a solid maintained at rest in a cyclic region in which a 
fluid is circulating irrotationally, and let K be the energy of the fluid, which 
will of course vary with the position of the solid. We will suppose the 
dimensions of the latter to be so small compared with the distances from the 
walls of the region that its position may be sufficiently given by point- 
co-ordinates (x, y, z). We have, then, for the components of the force exerted 
on it by the pressures of the fluid, 

X= _3Z Y _ZK Z = JK 

ox dy cz 

It remains to find, approximately, the form of this function K of oc, y, z. 
Let (u, v, w) be the velocity which the fluid would have at (x, y, z) if the 
solid were absent. If the solid were made to move with this velocity, and 
were of the same density as the surrounding fluid, the energy would be 
approximately the same as if the whole were fluid. It follows from Art. 141 
(19) that in this case the energy of the fluid would be ® + K, where 

2^ = Au 2 + Bv 2 + Cw 2 + 2A'vw + 2B' wu + 2C'uv, (2) 

by Art. 124, and that of the solid would be 

i p Q(u 2 + v 2 + w 2 ), (3) 

where Q is the volume displaced. The expression 

r ® + ipQ(u 2 + v 2 + w 2 ) + K (4) 

has therefore a constant value, viz. that of the energy of a fluid filling the 
region, and having the given circulations. This determines the form of K. 

Hence 

Y= ^ +i ^| (M2+w2+w2) ^ - (5) 

Z-.^ + ipQ^(u« + «• + «»). 

* G. I. Taylor, " The Forces on a Body placed in a Curved or Converging Stream of Fluid," 
Proc. Roy. Soc. cxx. 260 (1928). 



200 Motion of Solids through a Liquid [chap, vi 

Since the forces on the solid must depend only on the motion of the fluid 
in the immediate neighbourhood, these expressions are general, and inde- 
pendent of the special conception employed in their derivation. 

If the direction of the undisturbed stream, near the solid, be taken as the axis of x, the 
results simplify. Putting v = 0, w=0, we have 

/a ^3m _. dw ~,,dv) 

T={ ( A +P <; + B<g + o<! 

If, further, the stream is symmetrical with respect to the planes y=0, 2=0 we have 
' = 0, dujdz=0, and therefore also dv/dx = 0, dw/dx=0, on account of the assumed 
irrotational character. The symmetry also requires dw/dy=dv/dz=0. Hence 



-{ 



.(6) 



•(7) 



First suppose that one of the axes of permanent translation (Art. 124) coincides with 
the direction of the stream. Then C = 0, B' = 0, and 

X=(A+p£)/, Y=0, Z=0, (8) 

where / is the acceleration in the undisturbed stream. Thus if the solid is spherical, 
A = §7rpa 3 , Q=^vra 3 , X = 27rpa 3 /. For a circular cylinder, reckoning per unit length, 

A = 7rpa 2 , Q = 7rpa 2 , X = Zirpa 2 f. 
Next suppose merely that two of the axes of permanent translation lie in a plane with 
the direction of the stream. If the plane in question be that of xy we have A' = 0, B' = 0. 
If the stream is symmetrical about the axis of x, we have, further, 

dv _ aw__ 1 du 
dy ~ dz 2 dx' 
and the forces reduce to 

X = (A+ P G)/, Y=4C'/, Z=0 (9) 

In the case of a circular disk, 

A = |pa 3 cos 2 a, C'= -| pa 3 sin a cos a, $=0, 
where a is the angle which the stream makes with the axis of symmetry. In the two- 
dimensional case of the elliptic cylinder, 

A = n-p (b 2 cos 2 a + a 2 sin 2 a), C = irp (a 2 — b 2 ) sin a cos a, Q = nab, 
where a is now the inclination of the stream to the major axis*. 

The above theory has an interest in connection with the ' pressure-drop ' in a wind- 
channel, as used for measuring the drag of aircraft models. The stream of air converges 
slightly towards the fan at the forward end of the tunnel, and the increase of velocity 
implies a fall of pressure. We have then 

»/=-! oo) 

* These particular cases have been verified by direct calculation of the effect of the fluid 
pressures: Aeronautical Research Committee, R. and M. 1164 (1928). 



143-144] Kineto- Statics 201 

The preceding formulae shew that it would be incorrect to calculate the value of X from 
the observed pressure-gradient as if it were a statical question, in which case we should 
have "X. = pQf simply*. 

Some further interesting examples of Kineto-Statics (not reproduced in 
the present edition) have been discussed by Sir W. Thomson f, Kirchhoff {, 
and Boltzmann§. 

144. We here take leave of this branch of our subject. To avoid, as far 
as may be, the suspicion of vagueness which sometimes attaches to the use of 
' generalized co-ordinates,' an attempt has been made in this Chapter to put 
the question on as definite a basis as possible, even at the expense of some 
degree of prolixity in the methods. 

To some writers || the matter has presented itself as a much simpler one. 
The problems are brought at one stroke under the sway of the ordinary 
formulae of Dynamics by the imagined introduction of an infinite number of 
'ignored co-ordinates,' which would specify the configuration of the various 
particles of the fluid. The corresponding components of momentum are 
assumed all to vanish, with the exception (in the case of a cyclic region) of 
those which are represented by the circulations through the several apertures. 

From a physical point of view it is difficult to refuse assent to such a 
generalization, especially when it has formed the starting-point of all the 
development of this part of the subject; but it is at least legitimate, and 
from the hydrodynamical standpoint even desirable, that it should be verified 
a posteriori by independent, if more pedestrian, methods. 

Whichever procedure be accepted, the result is that the systems con- 
templated in this Chapter are found to comport themselves (so far as the 
'palpable' co-ordinates q 1} q 2 , ... q n are concerned) exactly like ordinary 
systems of finite freedom. The further development of the general theory 
belongs to Analytical Dynamics, and must accordingly be sought for in books 
and memoirs devoted to that subject. It may be worth while, however, to 
remark that the hydrodynamical systems afford extremely interesting and 
beautiful illustrations of the Principle of Least Action, the Reciprocal 
Theorems of Helmholtz, and other general dynamical theories. 

* G. I. Taylor, I.e. 

t "On the Forces experienced by Solids immersed in a Moving Liquid," Proc. R. S. Edin. 
1870 [Reprint, Art. xli.]. 

% I.e. ante p. 54. 

§ "Ueber die Druckkrafte welche auf Einge wirksam sind die in eine bewegte Flussigkeit 
tauchen," Crelle, lxxiii. (1871) [Wiss. Abh. i. 200]. 

|| See Thomson and Tait, and Larmor, 11. cit. ante p. 195. 



CHAPTER VII 

VORTEX MOTION 

145. Our investigations have thus far been confined for the most part 
to the case of irrotational motion. We now proceed to the study of rotational 
or 'vortex' motion. This subject was first investigated by Helmholtz*; other 
and simpler proofs of some of his theorems were afterwards given by Kelvin 
in the paper on vortex motion already cited in Chapter ill. 

We shall, throughout this Chapter, use the symbols f, 77, f to denote, as 
in Chapter III., the components of vorticity, viz. 

<._dw dv _du dw ^__dv _du . 

dy dz* dz dx' dx dy 

A line drawn from point to point so that its direction is everywhere that 
of the instantaneous axis of rotation of the fluid is called a 'vortex-line.' The 
differential equations of the system of vortex-lines are 

dx dy _ dz , g v 

T = 7 _ 7 () 

If through every point of a small closed curve we draw the corresponding 
vortex-line, we mark out a tube, which we call a ' vortex-tube.' The fluid 
contained within such a tube constitutes what is called a ' vortex-filament,' or 
simply a ' vortex.' 

Let ABC, A'B'C be any two circuits drawn on the surface of a vortex- 
tube and embracing it, and let A A' be a connecting line 
also drawn on the surface. Let us apply the theorem of 
Art. 32 to the circuit ABGAA'G'B'A A and the part of 
the surface of the tube bounded by it. Since 

/f +ra77 + nf=0 
at every point of this surface, the line-integral 

j(udx + vdy + wdz), 
taken round the circuit, must vanish ; i.e. in the notation of Art. 31 

I(ABCA) + I(AA') + I(A'C'B , A') + I(A'A) = 0, 
which reduces to I (ABC A) = 1 {A'B'C A'). 

Hence the circulation is the same in all circuits embracing the same vortex- 
tube. 

* "Ueber Integrate der hydrodynamischen Gleichungen welche den Wirbelbewegungen 
entsprechen," Crelle, lv. (1858) [Wiss. Abh. i. 101]. 




145-146] Persistence of Vortices 203 

Again, it appears from Art. 31 that the circulation round the boundary 
of any cross-section of the tube, made normal to its length, is coa, where 
<o, = (f 2 + 7? 2 + £ 2 )^, is the resultant vorticity of the fluid, and a the infinitely 
small area of the section. 

Combining these results we see that the product of the vorticity into the 
cross-section is the same at all points of a vortex. This product is conveniently 
taken as a measure of the ' strength ' of the vortex*. 

The foregoing proof is due to Kelvin ; the theorem itself was first given 
by Helmholtz, as a deduction from the relation 

which follows at once from the values of £, rj, J" given by (1). In fact writing, 
in Art. 42 (1), f, y, £ for U, V, W, respectively, we find 

fJ(K'+m + Od8~o, (4) 

where the integration extends over any closed surface lying wholly in the 
fluid. Applying this to the closed surface formed by two cross-sections of a 
vortex-tube and the part of the walls intercepted between them, we find 
ft>i0i = 0)20*2, where ©i, co 2 denote the vorticities at the sections oi, cr 2 , 
respectively. 

Kelvin's proof shews that the theorem is true even when f, 77, f are 
discontinuous (in which case there may be an abrupt bend at some point of a 
vortex), provided only that u, v, w are continuous. 

An important consequence of the above theorem is that a vortex-line 
cannot begin or end at any point in the interior of the fluid. Any vortex- 
lines which exist must either form closed curves, or else traverse the fluid, 
beginning and ending on its boundaries. Compare Art. 36. 

The theorem of Art. 32 (3) may now be enunciated as follows : The 
circulation in any circuit is equal to the sum of the strengths of all the 
vortices which it embraces. 

146. It was proved in Art. 33 that in a perfect fluid whose density is 
either uniform or a function of the pressure only, and which is subject to 
forces having a single-valued potential, the circulation in any circuit moving 
with the fluid is constant. 

Applying this theorem to a circuit embracing a vortex-tube we find that 
the strength of any vortex is constant. 

If we take at any instant a surface composed wholly of vortex-lines, 
the circulation in any circuit drawn on it is zero, by Art. 32, for we have 
1% + my + ft? = at every point of the surface. The preceding Art. shews 
that if the surface be now supposed to move with the fluid, the circulation 
will always be zero in any circuit drawn on it, and therefore the surface will 
* The circulation round a vortex being the most natural measure of its intensity. 



2(M Vortex Motion [chap, vii 

always consist of vortex-lines. Again, considering two such surfaces, it is 
plain that their intersection must always be a vortex-line, whence we derive 
the theorem that the vortex-lines move with the fluid. 

This remarkable theorem was first given by Helmholtz for the case of 
incompressibility ; the preceding proof, by Kelvin, shews that it holds for all 
fluids subject to the conditions above stated. 

The theorem that the circulation in any circuit moving with the fluid is 
invariable constitutes the sole and sufficient appeal to Dynamics which it 
is necessary to make in the investigations of this Chapter. It is based on 
the hypothesis of a continuous distribution of pressure, and (conversely) 
implies this. For if in any problem we have discovered functions u, v, w of 
x, y, z, t which satisfy the kinematical conditions, then, if this solution is to 
be also dynamically possible, the relation between the pressures about two 
moving particles A, B must be given by the formula (2) of Art. 33, viz. 

*£ + Cl-itf\ B = -^- \ B (udx + vdy + ivdz) (1) 

p \a ■Lftjj 

It is therefore necessary and sufficient that the expression on the right-hand 

side should be the same for all paths of integration (moving with the fluid) 

which can be drawn from A to B. This is secured if, and only if, the assumed 

values of u, v, w make the vortex-lines move with the fluid, and also make 

the strength of every vortex constant with respect to the time. 

It is easily seen that the argument is in no way impaired if the assumed 

values of u, v, w make f, rj, f discontinuous at certain surfaces, provided only 

that u, v, w are themselves everywhere continuous. 

On account of their historical interest, one or two independent proofs of the preceding 
theorems may be briefly indicated, and their mutual relations pointed out. 

Of these proofs, perhaps the most conclusive is based upon a slight generalization of 
some equations given originally by Cauchy in the introduction to his great memoir on 
Waves * and employed by him to demonstrate Lagrange's velocity-potential theorem. 

The equations (2) of Art. 15 yield, on elimination of the function % by cross-differentia- 
tion, 

du dx du dx dv dy dv dy dw dz dw dz dw dv 
db dc dc db db dc dc 36 cb dc dc db ~ 86 dc 
(where u, v, w have been written in place of dx/dt, dy/ot, dz/dt, respectively), with two 
symmetrical equations. If in these equations we replace the differential coefficients of 
u, v, w with respect to a, 6, c, by their values in terms of differential coefficients of the 
same quantities with respect to x, y, z, we obtain 

> d(y> g ) , %{z,x) / . d(x,y) _ ■) 
? 8(6,c) +?7 a(6,c) i "^ 9(6,c) *° 



^d(c,a)' i ' V d(c,a)' f ' i d(c,a) ? ' 

? 3(a,6)' r ^a(a,6)" + " J, 8(a, 6) i0 ' 
* I.e. ante p. 17. 



146] 



Helmholtz' Equations 



205 



If we multiply these by dx/da, dx/db, dx/dc, in order, and add, then, taking account of the 
Lagrangian equation of continuity (Art. 14 (1)) we deduce the first of the following three 
symmetrical equations : 

£ ^^, Vo dx . Co dx 



p p da p db p dc 



p da p db p dc 



,(3) 



p p da p 96 p dc 

In the particular case of an incompressible fluid (p=p ) these differ only in the use of 
the notation £, rj, £ from the equations given by Cauchy. They shew at once that if the 
initial values £ , ?7 , £ of the component vorticities vanish for any particle of the fluid, then 
£, 77, £ are always zero for that particle. This constitutes in fact Cauchy's proof of Lagrange's 
theorem. 

To interpret (3) in the general case, let us take at time t = 0& linear element coincident 
with a vortex-line, say 

da, 8b, fc«c&, - m - Co 



Po 



Po 



e — . 
Po' 



where e is infinitesimal. If we suppose this element to move with the fluid the equations 
(3) shew that its projections on the co-ordinate axes at any other time will be given by 

8x, By. 8z=e-, e-, e-, 
* P P P 

i.e. the element will still form part of a vortex-line, and its length (8s, say) will vary as 
ca/p, where o> is the resultant vorticity. But if a be the cross-section of a vortex -filament 
having 8s as axis, the product p<r8s is constant with regard to the time. Hence the strength 
oacr of the vortex is constant*. 

The proof given originally by Helmholtz depends on a system of three equations 
which, when generalized so as to apply to any fluid in which p is a function of p only, 
become t 

£ du rj du £ du 

pdx pdy pdz 

Dt\p) pfa + p*-' r ~ " 
D fC 



Dt\p) 



_i?2f! + 2: 

Dt \p) pdx p 



dy pdz 
p dz ' 



.(4) 



These may be obtained as follows. The dynamical equations of Art. 6 may be written, 
when a force-potential O exists, in the forms 

du 
dt' 



vc + wri= — iA- 

*> T ' cx 



Wt -wZ+uC= 






dw . dV 



•(5) 



provided 



H 



dp 



+^ 2 + Q, 



.(6) 



* See Nanson, Mess, of Math. hi. 120 (1874); Kirchhoff, Mechanik, c. xv. (1876) 
Papers, ii. 47 (1883). 
t Nanson, I.e. 



Stokes, 



206 Vortex Motion [chap, vii 

where q 2 = i£ l + v 2 + w 2 . From the second and third of these we obtain, eliminating % by 
cross-differentiation, 

Remembering the relation ^ + J- + ^=0, (7) 

and the equation of continuity 

Dp (du dv dw\ . . 

we easily deduce the first of equations (4). 

To interpret these equations we take, at time t, a linear element whose projections on 
the co-ordinate axes are 

to, ty, &»-*£, c2, c^, (9) 

P P P 
where e is infinitesimal. If this element be supposed to move with the fluid, the rate at 
which dx is increasing is equal to the difference of the values of u at the two ends, whence 

D8x £du ndu tdu 
Dt pdx pdy p oz 
It follows, by (4), that 

*(*-JH »(*-JH *(H)-° (10) 

Helmholtz concludes that if the relations (9) hold at time £, they will hold at time 
t + dt, and so on, continually. The inference is, however, not quite rigorous; it is in fact 
open to the criticisms which Stokes* directed against various defective proofs of Lagrange's 
velocity-potential theorem f. 

By way of establishing a connection with Kelvin's investigation we may notice that 
the equations (2) express that the circulation is constant in each of three infinitely small 
circuits initially perpendicular, respectively, to the three co-ordinate axes. Taking, for 
example, the circuit which initially bounded the rectangle 8b 8c, and denoting by A, B, C 
the areas of its projections at time t on the co-ordinate planes, we have 

4-|M»te, B=l%f>SbSc, C- d J*£»ic 

d (6, c) d (6, c) d (6, c) 

so that the first of the equations referred to is equivalent + to 

§A+ n B+£C=£ 8b8c (11) 

As an application of the equations (4) we may consider the motion of a liquid of uniform 
vorticity contained in a fixed ellipsoidal vessel §. The formulae 

u=qz — ry, v = rx—pz, w—py — qx (12) 

* I.e. ante p. 17. 

f It may be mentioned that, in the case of an incompressible fluid, equations somewhat 
similar to (4) had been established by Lagrange, Miscell. Taur. ii. (1760) [Oeuvres, i. 442]. The 
author is indebted for this reference, and for the above remark on Helmholtz' investigation, to 
Sir J. Larmor. Equations equivalent to those given by Lagrange were obtained independently by 
Stokes, I.e., and made the basis of a rigorous proof of the velocity-potential theorem. 

J Nanson, Mess, of Math. vii. 182 (1878). A similar interpretation of Helmholtz' equations 
was given by the author of this work in the Mess, of Math. vii. 41 (1877). 

Finally it may be noted that another proof of Lagrange's theorem, based on elementary 
dynamical principles, without special reference to the hydrokinetic equations, was indicated by 
Stokes, Camb. Trans, viii. [Papers, i. 113], and carried out by Kelvin in his paper on Vortex 
Motion. 

§ Cf. Voigt, "Beitrage zur Hydrodynamik," G'dtt. Nachr. 1891, p. 71; Tedone, Nuovo Cimento, 
xxxiii. (1893). The artifice in the text is taken from Poincare, "Sur la precession des corps 
deformables," Bull. Astr. 1910. 



146-147] Helmholtz' Equations 207 

obviously represent a uniform rotation of the fluid as a solid within a spherical boundary. 
Transforming the co-ordinates and the corresponding velocities by homogeneous strain we 

obtain the formulae 

u qz ry v _rx pz w _py qx . . 

a cbba ceo a 

as representing a certain motion within a fixed ellipsoidal boundary 

£+£+£-1 (") 



2 ' ft* ' c 



Thesemake «=g + £>, ,-£ + *)* f-g + |)r (15) 

Substituting in (4) we obtain 

(P+^ff-P 1 -*)^, (16) 

which may be written 

a 2 (6 2 +c 2 )^={6 2 (c 2 + a 2 )-c 2 (a 2 -|-6 2 )}^, (17) 

with two similar equations. We have here an identity as to form with Euler's equations 
of free motion of a solid about a fixed point. We easily deduce the integrals 

f 2 +p + §= const -> (18) 

6V£ 2 c 2 «V a 2 ^ 2 ^ , , im 

and £TT^+ 2 , 2 + o , ,, =const., (19) 

the former of which is a verification of one of Helmholtz' theorems, whilst the latter follows 
from the constancy of the energy. 

147. It is easily seen by the same kind of argument as in Art. 41 that 
no continuous irrotational motion is possible in an incompressible fluid filling 
infinite space, and subject to the condition that the velocity vanishes at 
infinity. This leads at once to the following theorem : 

The motion of a fluid which fills infinite space, and is at rest at infinity, 
is determinate when we know the values of the expansion (0, say) and of the 
component vorticities f , r) y f, at all points of the region. 

For, if possible, let there be two sets of values, u 1} v 1} w 1} and u 2 , tfo, w%, of 
the component velocities, each satisfying the equations 

du dv dw n 

T* + r y + is =0 > w 

dw dv _ £ du dw _ dv du _ y . 

dy~dz~~*' di~d^~ V} dx~dy~^ {> 

throughout infinite space, and vanishing at infinity. The quantities 

u' = u 1 — u 2 , v' = vx — v z , w , = w 1 — w 2 

will satisfy (1) and (2) with 6, f, 77, ? = 0, and will- vanish at infinity. Hence, 
in virtue of the result above stated, they will everywhere vanish, and there 
is only one possible motion satisfying the given conditions. 

In the same way we can shew that the motion of a fluid occupying any 
limited simply-connected region is determinate when we know the values of 



208 Vortex Motion [chap, vii 

the expansion, and of the component vorticities, at every point of the region, 
and the value of the normal velocity at every point of the boundary. In the 
case of an w-ply-connected region we must add to the above data the values 
of the circulations in n several independent circuits of the region. 

148. If, in the case of infinite space, the quantities 6, f, r), f all vanish 
beyond some finite distance of the origin, the complete determination of 
u, v, w in terms of them can be effected as follows *. 

The component velocities due to the expansion can be written down at 

once from Art. 56 (1), it being evident that the expansion 6' in an element 

8x'8y'8z' is equivalent to a simple source of strength O'Bx'Sy'Sz'. We thus 

obtain 

83> d® d<$> 

u = ~Tx> v = -fy> W = -Tz> (1) 

where <S> =~ [[[-dx'dy'dz', (2) 

r denoting the distance between the point (V, y\ z') at which the volume- 
element of the integral is situate and the point (x, y, z) at which the values 
of u, v, w are required, viz. 

r = {(x - x'f + (y- y'f + (z - z'f)$. 
The integration includes all parts of space at which 6 r differs from zero. 

To find the velocities due to the vortices, we note that when there is no 
expansion, the flux across any two open surfaces bounded by the same curve 
as edge will be the same, and will therefore be determined solely by the 
configuration of the edge. This suggests that the flux through any closed 
curve may be expressed as a line-integral taken round the curve, say 

J(Fdx+Gdy + Hdz) (3) 

On this hypothesis we should have, by the method of Art. 31, 

u=z dH_dG v = dF_dH w = ^_^l / 4) 

~" dy dz ' dz dx ' dx dy 

It is necessary and (as we have seen) sufficient that the functions F, G, H 
should satisfy 

djv_dv = l/dF + dG + dH\ V2F 
dy dz dx \ dx dy dz J 

together with two similar equations. They will in any case be indeterminate 
to the extent of three additive functions of the forms d%\dx, dx/dy, dx/dz, 
respectively, and we may, if we please, suppose x to be chosen so that 

dF dG dH ,-x 

^-- + ^- + ^- = 0, (5) 

dx dy dz 

* The investigation which follows is substantially that given by Helmholtz. The kinematical 
problem in question was first solved, in a slightly different manner, by Stokes, " On the Dynamical 
Theory of Diffraction," Camb. Trans, ix. (1849) [Papers, ii. 254...]. 



47r J J J r 



147-148] Velocities due to a Vortex-System 209 

in which case V 2 ^=-?, V 2 G = - V , ^ 2 H=-^ (6) 

Particular solutions of these equations are obtained by equating F, G, H to 
the potentials of distributions of matter whose volume-densities are f/47r, 
77/473-, ?/47r, respectively; thus 

? dx'dy'dz', G = ~ [ I [ £ dx'dy'dz', H = ^ [ [ j £ dx'dy'dz, 

(7) 

where the accents attached to f, 77, f are used to distinguish the values of 
these quantities at the point {x ', y', 2'). The integrations are to include, of 
course, all places where f, 97, f differ from zero. It remains to shew that these 
values of F, G, H do in fact satisfy (5). Since d/cx . r _1 = — d/dx'. r~\ the 
formulae (7) make 

dF dG dH 1 (ff/ H 3 1 ,81 w ai\, u ,,, 

The right-hand member vanishes, by a generalization of the theorem of 
Art. 42 (4)*, since 

3d? 3y 80 
everywhere, whilst Zf -f 77177 + ?if = 

at the surfaces of the vortices (where f, 77, J may be discontinuous), and 
f , 77, f vanish at infinity. 

The complete solution of our problem is obtained by superposition of the 
results contained in (1) and (4), viz. 

3# 9y 3s 

?J = _^ + ^_^ 

3y dz d% ' 

= _3<£ s^_M! 

82 3a? 3y 
where <£, F, G, # have the values given in (2) and (7). 

It may be added that the proviso that 0, f, 77, f should vanish beyond a 
certain distance from the origin is not absolutely essential. It is sufficient if 
the data be such that the integrals in (2) and (7), when taken over infinite 
space, are convergent. This will certainly be the case if 6, f, 77, £ are ultimately 
of the order R~ n , where R denotes distance from the origin, and n >3f. 

When the region occupied by the fluid is not unlimited, but is bounded 
(in whole or in part) by surfaces at which the normal velocity is given, and 
when further (in the case of an n-ip\y connected region) the value of the 
circulation in each of n independent circuits is prescribed, the problem may 

The singularity which occurs at the point r = is assumed to be treated here and elsewhere 
as in the theory of Attractions. The result is not affected, 
t Cf. Leathern, Cambridge Tracts, No. 1 (2nd ed.), p. 44. 



210 Vortex Motion [chap, vii 

by a similar analysis be reduced to one of irrotational motion, of the kind 
considered in Chapter III., and there proved to be determinate. This may be 
left to the reader, with the remark that if the vortices traverse the region, 
beginning and ending on the boundary, it is convenient to imagine them 
continued beyond it, or along the boundary, in such a manner that they form 
re-entrant filaments, and to make the integrals (7) refer to the complete 
system of vortices thus obtained. On this understanding the condition (5) 
will still be satisfied. 

There is an exact correspondence between the analytical relations above developed and 
certain formulae in Electro-magnetism. If, in the equations (1) and (2) of Art. 147, we 
write 

a, 3, y, p, u, % w, p 



for 




u, v t w, 0, |, Tj, C, 0, 


respectively, we obtain 




da 9/3 dy 

dx^dy^dz~ 9 ' 


dy 

w 


93 


9o dy _ 83 

dz dx ' dx 



.(9) 



da 

which are the fundamental relations of the theory referred to ; viz. o, 3, y are the compo- 
nents of magnetic force, u, v, w those of electric current, and p is the volume-density of the 
imaginary magnetic matter by which any magnetization present in the field may be repre- 
sented* Hence, the vortex- filaments correspond to electric circuits, the strengths of the 
vortices to the strengths of the currents in these circuits, sources and sinks to positive and 
negative magnetic poles, and, finally, fluid velocity to magnetic force t. 

The analogy will of course extend to all results deduced from the fundamental relations ; 
thus, in equations (8), <J> corresponds to the magnetic potential and F, G, H to the com- 
ponents of 'electro-magnetic momentum.' 

149. To interpret the result contained in Art. 148 (8), we may calculate 
the values of u, v, w due to an isolated re-entrant vortex-filament situate in 
an infinite mass of incompressible fluid which is at rest at infinity. 

Since = 0, we shall have <E> = 0. Again, to calculate the values of F, G, H, 
we may replace the volume-element Sx'Sy'Bz' by cr'Ss', where Ss r is an element 
of the length of the filament, and a its cross-section. Also 

, ,dx , r dy' , ,dz' 

Z=°>d?> v=03 d7> ?=G, 57' 

where a>' is the vorticity. Hence the formulae (7) of Art. 148 become 

F = —[— Q = JL\d]L h=—[— (1) 

47rJ r ' 47rJ r ' 47rJ r ' 

where k, = ©V, measures the strength of the vortex, and the integrals are to 
be taken along the whole length of the filament. 

* Cf . Maxwell, Electricity and Magnetism, Art. 607. The analogy has been improved by the 
adoption of the 'rational' system of electrical units advocated by Heaviside, Electrical Papers, 
London, 1892, i. 199. 

t This analogy was first pointed out by Helmholtz ; it has been extensively utilized by Kelvin 
in his papers on Electrostatics and Magnetism. 



148-150] Velocities due to an isolated Vortex 211 

Hence, by Art. 148 (4), we have 

with similar results for v, w. We thus find* 

tc [(dy z — z dz r y — y'\ ds' 



u = 



47r I \ds' r ds' 



?■ 



dz' x — x' dx' z — z'\ ds' 
ds' r ds' r ) r 2 



^k^^-^^i^.v (2) 



k Udx 



dx' y — y' dy' x — x'\ ds' 



w- , , , 2 

r ds r J it 



If 8u, 8v, 8w denote the parts of these expressions which involve the 
element 8s' of the filament, it appears that the resultant of 8u, Sv, 8w is 
perpendicular to the plane containing the direction of the vortex-line at 
(x, y', z') and the line r, and that its sense is that in which the point (x, y, z) 
would be carried if it were attached to a rigid body rotating with the fluid 
element at (x\ y', z'). For the magnitude of the resultant we have 

{(3^ + (8^ + (^|i=^ sin ^' ) (3) 

where x is tne angle which r makes with the vortex-line at (V, y', z'). 

With the change of symbols indicated in the preceding Art. this result becomes identical 
with the law of action of an electric current on a magnetic pole +. 

Velocity -Potential due to a Vortex. 

150. At points external to the vortices there exists a velocity-potential, 
whose value may be obtained as follows. Taking for shortness the case of a 
single re-entrant vortex, we have, from the preceding Art., in the case of an 
incompressible fluid, 

"'iiWr-V-bW (1) 

By Stokes' Theorem (Art. 32 (4)) we can replace a line-integral extending 
round a closed curve by a surface-integral taken over any surface bounded 
by that curve; viz. we have, with a slight change of notation, 

j W+w+ ^,=j|{«(f-g) + .(g-g) + .(i-i)}-- 

If we put P = 0, Q = S-,~, R = -^-,- 

oz r : "-' 

we find 

dJl_°Q = d 2 il_ dP dR = d 2 1 

dy' dz' dx'*r" dz' dx ~ dx'dy' r' ' 

* These are equivalent to the forms obtained by Stokes, I.e. ante p. 208. 

f Ampere, Theorie mathematique des phenomenes electro-dynamiques, Paris, 1826. 



d 

dy' 


1 






dQ 

dx' 


dP 


a 2 

dx'dz' 


1 



212 Vortex Motion [chap, vii 

so that (1) may be written 



U = £r\\{ 



, 3 3 3 \ d 1 7af 



Hence, and by similar reasoning, we have, since d/daf. r~ x = — d/dx . r -1 , 

— 1> — |. —I < 2 > 

where * = £ JJ(^ + ™|> + «^) ^ (3) 

Here I, m, w denote the direction-cosines of the normal to the element 8S' of 
a surface bounded by the vortex-filament. 
The formula (3) may be otherwise written 

cos^ 



*=£// 



dsr, (4) 



where S- denotes the angle between r and the normal (I, m, n). Since 
cos ^dS'/r 2 measures the elementary solid angle subtended by 8S' at (x, y, z), 
we see that the velocity-potential at any point, due to a single re-entrant 
vortex, is equal to the product of k/^tt into the solid angle which a surface 
bounded by the vortex subtends at that point. 

Since this solid angle changes by 4>tt when the point in question describes 
a circuit embracing the vortex, we verify that the value of <f> given by (4) is 
cyclic, the cyclic constant being tc. Cf. Art. 145. 

It may be noticed that the expression in (4) is equal to the flux (in the 
negative direction) through the aperture of the vortex, due to a point-source 
of strength k at the point (a?, y, z). 

Comparing (4) with Art. 56 (4) we see that a vortex is, in a sense, 
equivalent to a uniform distribution of double sources over any surface 
bounded by it. The axes of the double sources must be supposed to be 
everywhere normal to the surface, and the density of the distribution to be 
equal to the strength of the vortex. It is here assumed that the relation 
between the positive direction of the normal and the positive direction of the 
axis of the vortex-filament is of the 'right-handed' type. See Art. 31. 

Conversely, it may be shewn that any distribution of double sources over 
a closed surface, the axes being directed along the normals, may be replaced 
by a system of closed vortex-filaments lying in the surface*. The same thing 
will appear independently from the investigation of the next Art. 

Vortex-Sheets. 

151. We have so far assumed u, v, w to be continuous. We may now 
shew how cases where surfaces of discontinuity present themselves may be 
brought within the scope of our theorems. 

* Cf. Maxwell, Electricity and Magnetism, Arts. 485, 652. 



i50-i5i] Vortex-Sheets 213 

The case of a discontinuity in the normal velocity alone has already been 
treated in Art. 58. If u, v, w denote the component velocities on one side, 
and u\ v\ w' those on the other, it was found that the circumstances could 
be represented by imagining a distribution of simple sources, with surface- 
density 

l{u'—u) + m (v f — v) + n {w' — w), 

where I, ra, n denote the direction-cosines of the normal drawn towards the 
side to which the accents refer. 

Let us next consider the case where the tangential velocity (only) is dis- 
continuous, so that 

l(u'-u) + m(v'-v)+n(w'-<w) = (1) 

We will suppose that the lines of relative motion, which are defined by the 
differential equations 

dx dy dz (i> 



are traced on the surface, and that the system of orthogonal trajectories to 
these lines is also drawn. Let PQ, P'Q' be linear elements drawn close to 
the surface, on the two sides, parallel to a line of the system (2), and let PP' 
and QQ' be normal to the surface and infinitely small in comparison with PQ 
or P'Q'. The circulation in the circuit P'Q'QP will then be equal to (q'—q) PQ, 
where q, q' denote the absolute velocities on the two sides. This is the same 
as if the position of the surface were occupied by an infinitely thin stratum 
of vortices, the orthogonal trajectories above-mentioned being the vortex- 
lines, and the vorticity co and the (variable) thickness 8n of the stratum 
being connected by the relation 

(o8n = q' — q (3) 

The same result follows from a consideration of the discontinuities which 
occur in the values of u, v, w as determined by the formulae (4) and (7) of 
Art. 148, when we apply these to the case of a stratum of thickness 8n within 
which £, t), f are infinite, but so that £8n, rjBn, ^8n are finite*. 

It was shewn in Arts. 147, 148 that any continuous motion of a fluid 
filling infinite space, and at rest at infinity, may be regarded as due to a 
suitable arrangement of sources and vortices distributed with finite density. 
We have now seen how by considerations of continuity we can pass to the 
case where the sources and vortices are distributed with infinite volume- 
density, but infinite surface-density, over surfaces. In particular, we may take 
the case where the infinite fluid in question is incompressible, and is divided 
into two portions by a closed surface over which the normal velocity is con- 
tinuous, but the tangential velocity discontinuous, as in Art. 58 (12). This is 

* Helmholtz, I.e. ante p. 202. 



214 Vortex Motion [chap, vii 

equivalent to a vortex-sheet; and we infer that every continuous irrotational 
motion, whether cyclic or not, of an incompressible substance occupying any 
region whatever, may be regarded as due to a certain distribution of vortices 
over the boundaries which separate it from the rest of infinite space. In the 
case of a region extending to infinity, the distribution is confined to the finite 
portion of the boundary, provided the fluid be at rest at infinity. 

This theorem is complementary to the results obtained in Art. 58. 

The foregoing conclusions may be illustrated by means of the results of Art. 91. Thus 
when a normal velocity S n was prescribed over the sphere r=a, the values of the velocity- 
potential for the internal and external space were found to be 



♦-;©"*. -♦~ 5 j I gT*. 



respectively. Hence if 8e be the angle which a linear element drawn on the surface 
subtends at the centre, the relative velocity estimated in the direction of this element 
will be 

2n + l &S; 

n(n + l) de ' 

The resultant relative velocity is therefore tangential to the surface, and perpendicular to 
the contour lines (S n = const.) of the surface-harmonic S n , which are therefore the vortex- 
lines. 

For example, if we have a thin spherical shell filled with and surrounded by liquid, 
moving as in Art. 92 parallel to the axis of x, the motion of the fluid, whether internal or 
external, will be that due to a system of vortices arranged in parallel circles on the sphere ; 
the strength of an elementary vortex being proportional to the projection, on the axis of %, 
of the breadth of the corresponding zone of the surface*. 

Impulse and Energy of a Vortex-System. 

152. The following investigations relate to the case of a vortex-system of 
finite dimensions in an incompressible fluid which fills infinite space and is 
at rest at infinity. 

The problem of finding a distribution of impulsive force (X', Y', Z') per 
unit mass which would generate the actual motion (u, v, w) instantaneously 
from rest is to some extent indeterminate, but a sufficient solution for our 
purpose may be obtained as follows. 

We imagine a simply-connected surface S to be drawn enclosing all the 
vortices. We denote by </> the single-valued velocity-potential which obtains 
outside S, and by <j> x that solution of V 2 <£ = which is finite throughout the 
interior of S, and is continuous with <f> at this surface. In other words, </>i is 
the velocity -potential of the motion which would be produced within $ by the 
application of impulsive pressures p<f> over the surface. If we now assume 

*-+£. *--+£• z '=™ + t « 

* The same statements hold also for an ellipsoidal shell moving parallel to one of its principal 
axes. See Art. 114. 



151-152] Impulse of a Vortex-System 215 

at internal points, and 

Z'=0, 7'=0, Z'=0 (2) 

at external points, it is evident on reference to Art. 11 that these forces would 
in fact generate the actual motion instantaneously from rest, the distribution 
of impulsive pressure being given by p<f> at externa], and pfa at internal, points. 
The forces are discontinuous at the surface, but the discontinuity is only in 
the normal component, the tangential components vanishing just inside and 
just outside owing to the continuity of </> with fa. Hence if (I, m, n) be the 
direction-cosines of the inward normal, we should have 

rf-nF'=0, nX'-lZ'=0, ZF'-mX'=0, (3) 

at points just inside the surface. 

Now if we integrate over the volume enclosed by S we have 

jjj(yr- •») dxdydz=\\\\y g - g) -. g - g)} dxdydz 

= -fJ{y(lY , -mX , )-z(nX , -lZ , )}dS + 2jfJX , dxdydz, (4) 

where the surface-integral vanishes in virtue of (3). 
Again 

- jj}(2/ 2 + * 2 ) f dxdydz = - JJJ(y»+ *■) (g' - 1) d«4y<b 

= tf(y*+z 2 )(mZ'--nY')dS + 2fff{yZ'-zY')dxdydz, (5) 

where the surface-integral vanishes as before. 

We thus obtain for the force- and couple-resultants of the impulse of the 
vortex-system the expressions 

•P = ipfff(yS ~ z v) dxdydz, L = -\pjjj(y 2 + z 2 ) ^dxdydz; 
Q=ipJJK*Z-*S) dxdydz, M=-yfff(z*+x*) v dxdydz,r...(6) 
R = ipIII( x V - y%) dxdydz, N = -ipffj(x 2 + y 2 ) {dxdydz. 
To apply these to the case of a single re-entrant vortex-filament of infinitely 
small section a, we replace the volume element by a 8s, and write 

j. dx dy ., dz ,,_. 

i = ( °ds> V = ( °ts> ^ =( °ds (0 

Hence P = ipco<rf(ydz -zdy) = Kpffl'dS', (8) 

L = -ycoaj(y 2 + z 2 )dx = -fcpjf(m , z-n , y)dS', (9) 

with similar formulae. The line-integrals are supposed to be taken along the 
filament, and the surface-integrals over a barrier bounded by it, and V , m', n 
are the direction-cosines of the normal to an element 8S' of the barrier. The 



Z-%-W + F(t) (2) 



216 Vortex Motion [chap, vii 

identities of the different forms follow from Stokes' Theorem. We have also 
written k for cocr, i.e. tc is the circulation round the filament*. 

The whole investigation has reference of course to the instantaneous state 
of the system, but it may be recalled that, when no extraneous forces act, the 
impulse is, by the argument of Art. 119, constant in every respect. 

153. Let us next consider the energy of the vortex-system. It is easily 
proved that under the circumstances presupposed, and in the absence of 
extraneous forces, this energy will be constant. For if T be the energy 
of the fluid bounded by any closed surface S, we have, putting V=0 in 
Art. 10 (5), 

DT 

-jr- = Jf(lu + mv + nw) pdS (1) 

If the surface S enclose all the vortices, we may put 

p _ d(j> 

P 

and it easily follows from Art. 150 (4) that at a great distance R from the 
vortices p will be finite, and lu + mv -f nw of the order R~ z , whilst when the 
surface S is taken wholly at infinity, the elements 8S vary as R 2 . Hence, 
ultimately, the right-hand side of (1) vanishes, and we have 

T= const (3) 

We proceed to investigate one or two important kinematical expressions 
for T, still confining ourselves, for simplicity, to the case where the fluid (sup- 
posed incompressible) extends to infinity, and is at rest there, all the vortices 
being within a finite distance of the origin. 

The first of these expressions is indicated by the electro-magnetic analogy 
pointed out in Art. 148. Since = 0, and therefore <I> = 0, we have 

2T=pfff(u 2 + v 2 + w 2 ) dxdydz 

= p \\\{ u (f - S) + v (£- s) + w (£- !)} dxdydz > 

by Art. 148 (4). The last member may be replaced by the sum of a surface- 
integral 

pff{F(mw — nv)+ G (nu — Iw) + H{lv- mu)} dS, 

and a volume-integral 

* The expressions (8) and (9) were obtained by elementary reasoning by J. J. Thomson, On 
the Motion of Vortex Rings (Adams Prize Essay), London, 1883, pp. 5, 6, and the formulae (6) 
deduced from them, with, however, the opposite signs in the case of L, M, N. The correction is 
due to Mr Welsh. 

An interesting test of the formulae as they now stand is afforded by the case of a spherical 
mass rotating as if solid and surrounded by fluid at rest, provided we take into account the 
spherical vortex-sheet which represents the discontinuity of velocity. 



152-153] Energy of a Vortex- System 217 

At points of the infinitely distant boundary, F, G, H are ultimately of the 

order R~ 2 . and u, v, w of the order R~ z , so that the surface-integral vanishes, 

and we have 

T=i P JfJ(FS+Q v + H0dxdydz (4) 

or, substituting the values of F, 0, H from Art. 148 (7), 

T " t JJJjJJ ~ "r' + rr *"****'<&*' (*) 

where each volume- integration extends over the whole space occupied by the 

vortices. 

A slightly different form may be given to this expression as follows. 

Regarding the vortex-system as made up of filaments, let 8s, 8s' be elements 

of length of any two filaments, cr, a the corresponding cross-sections, and co, co' 

the corresponding vorticities. The elements of volume may be taken to be 

cr8s and cr'8s\ respectively, so that the expression following the integral signs 

in (5) is equivalent to 

C0S€ a„ ' '*«' 
. coats . co <t 6s , 

r 

where e is the angle between 8s and 8s' . If we put co<t — k, co'a' — k , we have 

T " ir XKK ' \\ ^ dsdS ' (6) 

where the double integral is to be taken along the axes of the filaments, and 
the summation 2 includes (once only) every pair of filaments which are present. 

The factor of p in (6) is identical with the expression for the energy of a system of 
electric currents flowing along conductors coincident in position with the vortex-filaments, 
with strengths k, k', ... respectively*. The above investigation is in fact merely an inversion 
of the argument given in treatises on Electro-magnetism, whereby it is proved that 

^2ii' f f^ dsds' = h f f f(a*+p 2 + y 2 )dxdydz, 

where i, % denote the strengths of the currents in the linear conductors whose elements are 
denoted by 8s, 8s' , and a, 3, y are the components of magnetic force at any point of the field. 

The theorem of this Art. is purely kinematical, and rests solely on the assumption that 
the functions u, v, w satisfy the equation of continuity, 

du dv cw _ Ci 

dx dy dz ' 

throughout infinite space, and vanish at infinity. It can therefore by an easy generaliza- 
tion be extended to a case considered in Art. 144, where a liquid is supposed to circulate 
irrotationally through apertures in fixed solids, the values of u, v, w being now taken to be 
zero at all points of space not occupied by the fluid. The investigation of Art. 151 shews 
that the distribution of velocity thus obtained may be regarded as due to a system of 
vortex-sheets coincident with the bounding surfaces. The energy of this system will be 
given by an obvious adaptation of the formula (6) above, and will therefore be proportional 
to that of the corresponding system of electric current-sheets. This proves a statement 
made by anticipation in Art. 144. 

* The ' rational ' system of electrical units being understood ; see ante p. 210. 



218 Vortex Motion [chap, vii 

Under the circumstances stated at the beginning of Art. 152, we have 
another useful expression for T\ viz. 

T— pfff{u(y£— zrj) + v (zg — x£) -hw(xr) — y%)} dxdydz* (7) 

To verify this, we take the right-hand member, and transform it by the process 
already so often employed, omitting the surface-integrals for the same reason 
as in the preceding Art. The first of the three terms gives 

p S\S u [ y (£ - D - * £ - £)} dxdydz 

~ ~~ P ) ( v y "** wz } a — u \ d°°dydz. 



dx j 

Transforming the remaining terms in the same way, adding, and making use 
of the equation of continuity, we obtain 

p I ( w 2 ■+- v 2 + w 2 + xu x- + yv =- + zw ~- J dxdydz, 

or, finally, on again transforming the last three terms, 

ipfff(u 2 + v 2 + w 2 ) dx dydz. 
In the case of a finite region the surface-integrals must be retained"]-. 
This involves the addition to the right-hand side of (7) of the term 

pfS {(^ u + mv + nw) {xu + yv + zw) — J (Ix 4- my + nz) q 2 \ dS, (8) 

where q 2 = u 2 + v 2 + w 2 . This simplifies in the case of a fixed boundary. 

The value of the expression (7) must be unaltered by any displacement of 
the origin of co-ordinates. Hence we must have 
///(«?— wrf) dx dy dz — 0, ///( w% — u%) dx dy dz = 0, JJf(u7j — v%) dx dy dz = 0. 

(9) 

These equations, which may easily be verified by partial integration, follow also from 
the consideration that if there are no extraneous forces the components of the impulse 
parallel to the co-ordinate axes must be constant. Thus, taking first the case of a fluid 
enclosed in a fixed envelope of finite size, we have, in the notation of Art. 152, 

P=pjjfudxdydz-pjfl<pdS, (10) 

if cp denote the velocity-potential near the envelope, where the motion is irrotational. 

Hence d ± =p j j J " *? dxdydz- P j fl^dS 

= ~ p l I \% dxd y dz + p\ I {(vC-wt)) dxdydz- pi (l^dS, (11) 

by Art. 146 (5). The first and third terms of this cancel, since at the envelope we have 
x' = dcp/dt, by Art. 20 (4) and Art. 1 46 (6). Hence for any re-entrant system of vortices 
enclosed in a fixed vessel, we have 

dP 

-^=pjjj(v£-wri) dxdydz, (12) 

witb two similar equations. It has been proved in Art. 119 that if the containing vessel 
be infinitely large, and infinitely distant from the vortices, P is constant. This gives the 
first of equations (9). 

* Motion of Fluids, Art. 136 (1879). 
t J. J. Thomson, I.e. ante p. 216. 



153-154] Rectilinear Vortices 219 

Conversely from (9), established otherwise, we could infer the constancy of the com- 
ponents P, Q, R of the impulse*. 

Rectilinear Vortices. 

154. When the motion is in two dimensions x, y we have w = 0, whilst 
u, v are functions of x, y, only. Hence f = 0, rj = 0, so that the vortex-lines 
are straight lines parallel to *. The theory then takes a very simple form. 

The formulae (8) of Art. 148 are now replaced by 

dx dy ' dy dx ' 

the functions <f>, yfr being subject to the equations 

V^ — 0, W-C (2) 

where V ^ 2 = ^ + ^' 

and to the proper boundary-conditions. 

In the case of an incompressible fluid, to which we will now confine our- 
selves, we have >' 

U = -fy> V = Tx> (3) 

where ty is the stream-function of Art. 59. It is known from the Theory of 
Attractions that the solution of 

V 1 2 ^ = ?, (4) 

X being a given function of x y y, is 

* = ^//?'logr<fe'A/ + *o. (5) 

where f denotes the value of f at the point (x' } y'), and r stands for 

{( x - x >f + (y-y'f}l. 
The 'complementary function' ^ may he any solution of 

V! 2 ^ = 0; (6) 

it enables us to satisfy the boundary-conditions. 

In the case of an unlimited mass of liquid, at rest at infinity, -^ is constant. 
The formulae (3) and (5) then give 

»=-^[[?'^W, **ki\r m ^*** < 7) 

Hence a vortex-filament whose co-ordinates are x', y' and whose strength is 
k contributes to the motion at (x, y) a velocity whose components are 

k v — y' , tc x — x' 

2tt r 2 27r r 2 

This velocity is perpendicular to the line joining the points (x, y), (x' } y'), and 
its amount is KJIirr. 

* J. J. Thomson, I.e. 



220 Vortex Motion [chap, vii 

Let us calculate the integrals ffu^dxdy, and ffvgdxdy, where the integra- 
tions include all portions of the plane xy for which f does not vanish. We 
have 



u Zdcody = ~ jjjfe' V -^/ dxdydx'dy', 



where each double integration includes the sections of all the vortices. Now, 
corresponding to any term 

&' y -^fdxdydx'dy' 

of this result, we have another 

KK' 2 dxdydx'dy', 
and these two neutralize each other. Hence, and by similar reasoning, 

ffu£dxdy = 0, Jfv£dxdy = (8) 

If as before we denote the strength of a vortex by k, these results may be 

written 

2fcu = 0, Xkv = (9) 

Since the strength of each vortex is constant with regard to the time, the 
equations (9) express that the point whose co-ordinates are 

-S- *-g ™ 

is fixed throughout the motion. 

This point, which coincides with the centre of inertia of a film of matter 
distributed over the plane xy with the surface-density f, may be called the 
'centre' of the system of vortices, and the straight line parallel to z of which 
it is the projection may be called the 'axis' of the system. If Xk — } the 
centre is at infinity, or else indeterminate. 

155. Some interesting examples are furnished by the case of one or more 
isolated vortices of infinitely small section. Thus : 

1°. Let us suppose that we have only one vortex-filament present, and that 
the vorticity f has the same sign throughout its infinitely small section. Its 
centre, as just defined, will lie either within the substance of the filament, or 
infinitely close to it. Since this centre remains at rest, the filament as a whole 
will be stationary, though its parts may experience relative motions, and its 
centre will not necessarily lie always in the same element of fluid. Any particle 
at a finite distance r from the centre of the filament will describe a circle 
about the latter as axis, wftth constant velocity kJ^ttv. The region external to 
the vortex is doubly-connected; and the circulation in any (simple) circuit 
embracing it is of course k. The irrotational motion of the surrounding fluid 
is the same as in Art. 27 (2). 



]54 



-155] 



Vortex-Pair 



221 



2°. Next suppose that we have two vortices, of strengths k\ } tc 2 , respectively. 
Let A , B be their centres, the centre of the system. The motion of each 
filament as a whole is entirely due to the other, and is therefore always per- 
pendicular to AB. Hence the two filaments remain always at the same distance 
from one another, aod rotate with constant angular velocity about 0, which 
is fixed. This angular velocity is easily found; we have only to divide the 
velocity of A (say), viz. # a /(27r. AB), by the distance AO, where 

/e 2 



AO 



/C1 + /C2 



AB, 



and so obtain 

Lit . A. if 

If «i, k 2 be of the same sign, i.e. if the directions of rotation in the two 
vortices be the same, lies between A and B; but if the rotations be of 
opposite signs, lies in AB, or B A, produced. 

If k±= ■ — k 2 , is at infinity; but it is easily seen that A, B move with 
equal velocities k^(2it . AB) at right angles to AB, which remains fixed in 
direction. Such a combination of two equal and opposite vortices may be 
called a 'vortex-pair.' It is the two-dimensional analogue of a circular vortex- 
ring (Art. 160), and exhibits many of the properties of the latter. 

The stream-lines of a vortex-pair form a system of coaxal circles, as shewn 
on p. 67, the vortices being at the limiting points (+ a, 0). To find the relative 




stream-lines, we superpose a general velocity equal and opposite to that of the 
vortices, and obtain, for the relative stream-function, 



r 2tt V2a T 



'<;)■ 



.(i) 



in the notation of Art. 64, 2°. The figure (which is turned through 90° for 
convenience) shews a few of the lines. The line yfr — consists partly of the 
axis of y, and partly of an oval surrounding both vortices. 



222 Vortex Motion [chap, vii 

It is plain that the particular portion of fluid enclosed within this oval 
accompanies the vortex-pair in its career, the motion at external points being 
exactly that which would be produced by a rigid cylinder having the same 
boundary; cf. Art. 71. The semi-axes of the oval are 2'09 a and 1*73 a, 
approximately *. 

A difficulty is sometimes felt, in this as in the analogous instance of a vortex-ring, 
in understanding why the vortices should not be stationary. If in the figure on p. 70 
the filaments were replaced by solid cylinders of small circular section, the latter might 
indeed remain at rest, provided they were rigidly connected by some contrivance which 
did not interfere with the motion of the fluid ; but in the absence of such a connection 
they would in the first instance be attracted towards one another, on the principle 
explained in Art. 23. This attraction is however neutralized if we superpose a general 
velocity V of suitable amount in the direction opposite to the cyclic motion half-way 
between the cylinders. To find V, we remark that the fluid velocities at the two points 
(a±c, 0), where c is small, will be approximately equal in absolute magnitude, provided 

V+— — = — +— - V 

2irC 4-rra 2irc Ana ' 



where k is the circulation. Hence 

V 



Ana 



which is exactly the velocity of translation of the vortex-pair, in the original form of the 
problem t. 

Since the velocity of the fluid at all points of the plane of symmetry is 
wholly tangential, we may suppose this plane to form a rigid boundary of the 
fluid on either side of it, and so obtain the case of a single rectilinear vortex 
in the neighbourhood of a fixed plane wall to which it is parallel. The filament 
moves parallel to the plane with the velocity KJ^rrh, where h is the distance 
from the wall. 

Again, since the stream-lines are circles, we can also derive the solution 
of the case where we have a single vortex-filament in a space bounded, either 
internally or externally, by a fixed circular cylinder. 

Thus, in the figure, let EPD be the section of the cylinder, A the position of the vortex 
(supposed in this case external), and let B be the 'image' of A with respect to the circle 
EPD, viz. C being the centre, let 

CB.CA=c 2 , 

where c is the radius of the circle. If P be any point on 
the circle, we have 

AP _AE _ AD_ 

BP~ BE~ BD~° ' 

so that the circle occupies the position of a stream-line due 

to a vortex-pair at A, B. Since the motion of the vortex A would be perpendicular to AB, 

* Cf. Sir W. Thomson, "On Vortex Atoms," Phil. Mag. (4), xxxiv. 20 (1867) [Papers, iv. 1]; 
and Eiecke, Gott. Nachr. 1888, where paths of fluid particles are also delineated. 

t A more exact investigation is given by Hicks, "On the Condition of Steady Motion of Two 
Cylinders in a Fluid," Quart. Journ. Math. xvii. 194 (1881). 




155] 



Special Cases 



223 



it is plain that all the conditions of the problem will be satisfied if we suppose A to 
describe a circle about the axis of the cylinder with the constant velocity 

< k.CA 

2n.AB~ $tt(CA 2 -c 2 )' 
where k denotes the strength of A. 

In the same way a single vortex of strength k, situated inside a fixed circular cylinder, 
say at B, would describe a circle with constant velocity 

k.CB 
2tt(c 2 -CB 2 )' 
It is to be noticed, however* that in the case of the external vortex the motion is not 
completely determinate unless, in addition to the strength k, the value of the circulation 
in a circuit embracing the cylinder (but not the vortex) is prescribed. In the above 
solution, this circulation is that due to the vortex-image at B and is — k. This may be 
annulled by the superposition of an additional vortex + k at (7, in which case we have, for 
the velocity of A, 

K.CA K KC 2 

~ 2rr(CA 2 ~c 2 ) + 27r.CA 2tt . CA (CA 2 -C 2 ) ' 

For a prescribed circulation k we must add to this the term k'\2tt . CA. 

L. Foppl t, using the method of images, has investigated the case of a cylinder advancing 
through fluid with velocity U, and followed by a vortex-pair symmetrically situated with 
respect to the line of advance of the centre. It appears that the vortices can maintain 
their position relative to the cylinder provided they lie on the curve 

2ry = r 2 — a 2 , 
and that the strengths of the vortices corresponding to a given position on this curve are 



+ 2 



*0-3) 



He finds, however, that the arrangement is unstable for anti-symmetrical disturbances. 

Some paths of vortices in a stream past a cylindrical obstacle (with circulation) have 
been traced by Walton J. The path of a vortex in a semicircular region is investigated by 
K. De § by Routh's method referred to on p. 224. 

3°. If we have four parallel rectilinear vortices whose centres form a 
rectangle ABB' A', the strengths being k for the vortices A' } B, and — k for 






the vortices A, B\ it is evident that the centres will always form a rectangle 
* F. A. Tarleton, "On a Problem in Vortex Motion," Proc. R. I. A. December 12, 1892. 
f " Wirbelbewegung hinter einem Kreiszylinder," Sitzb. d. k. bdyr. Akad. d. Wiss. 1913. 
| Proc. R. I. Acad, xxxviii. A (1928). 
§ Bull, of the Calcutta Math. Soc. xxi. 197 (1929). 



224 Vortex Motion [chap, vii 

Further, the various rotations having the directions indicated in the figure, 
we see that the effect of the presence of the pair A, A' on B, B' is to separate 
them, and at the same time to diminish their velocity perpendicular to the 
line joining them. The planes which bisect AB, AA' at right angles may 
(either or both) be taken as fixed rigid boundaries. We thus get the case 
where a pair of vortices, of equal and opposite strengths, move towards (or 
from) a plane wall, or where a single vortex moves in the angle between two 
perpendicular walls. 

If x, y be the co-ordinates of the vortex A relative to the planes of symmetry, we 
readily find 

. <_ #2 -_ji_ y^_ /on 

X ~ 4n-yr*> V ~^' xr 2 " { } 

where r 2 — x 2 -\~y 2 . By division we obtain the differential equation of the path, viz. 

x 3 + y* ' 
whence a 2 (x 2 +y 2 ) = 4x 2 y 2 , 

a being an arbitrary constant, or, transforming to polar co-ordinates, 

-sk (3) 

Also since x y-y%=-r~> 

the vortex moves as if under a centre of force at the origin. This force is repulsive, and 
its law is that of the inverse cube*. 

156. If we write, as in Chapter IV., 

z = oc + iy, w = (f> + iyjr, (1) 

the potential- and stream -functions due to an infinite row of equidistant 
vortices, each of strength k, whose co-ordinates are 

(0,0), (±a, 0), (±2o, 0), ..., 

will be given by the formula 

ifc , . irz . 

w = ^logsm-; (2) 

cf. Art. 64, 4°. This makes 

dw %k , irz /ox 

u — iv— — 5-= — s- cot — , (6) 

dz 2a a 

whence 

k sinh {1iry\a) ___ k sin(27r#/a) 

2a cosh (27n//a) — cos (2wx/a) ' 2a cosh (2iry/a) — cos (27nc/a) ' 

(4) 

* Greenhill, "On Plane Vortex- Motion," Quart. Journ. Math. xv. 10 (1878); Grobli, Die 
Bewegung paralleler geradliniger Wirbelfdden, Zurich, 1877. These papers contain other in- 
teresting examples of rectilinear vortex-systems. The case of a system of equal and parallel 
vortices whose intersections with the plane xy are the angular points of a regular polygon was 
treated by J. J. Thomson in his Motion of Vortex Rings, pp. 94.... He finds that the configura- 
tion is stable if, and only if, the number of vortices does not exceed six. For some further 
references as to special problems see Hicks, Brit. Ass. Rep. 1882, pp. 41...; Love, I.e. ante p. 192. 

An ingenious method of transforming plane problems in vortex-motion was given by Kouth, 
"Some Applications of Conjugate Functions," Proc. Lond. Math. Soc. xii. 73 (1881). 



155-156] Rows of Vortices 225 

These expressions make w= + \ic\a, v = 0, for y = ± oo ; the row of vortices is 
in fact, as regards distant points, equivalent to a vortex-sheet of uniform 
strength x/a (Art. 151). 




The diagram shews the arrangement of the stream-lines. 

It follows easily that if there are two parallel rows of equidistant vortices, 
symmetrical with respect to the plane y = 0, the strengths being k for the 
upper and — k for the lower row, as indicated on the next page, the whole 
system will advance with a uniform velocity 

£7 = ^coth— , (5) 

where b is the distance between the two rows. The mean velocity in the plane 
of symmetry is KJa. The velocity at a distance outside the two rows tends to 
the limit 0. 

If the arrangement be modified so that each vortex in one row is opposite 
the centre of the interval between two consecutive vortices in the other row, 
as shewn on p. 228, the general velocity of advance is 

F =^ tanh ? w 

The mean velocity in the medial plane is again k/cl. 

The stability of these various arrangements has been discussed by von Karman*. 
Taking first the case of the single row, let us suppose the vortex whose undisturbed 
co-ordinates are (ma, 0) to be displaced to the point (ma + x m , y m ). The formulae of 
Art. 154 give, for the motion of the vortex initially at the origin, 

dxQ = k yo-y m dy Q ^ < x -x m -ma 
dt 2rr m rj ' dt 2tt m rj ' { '> 

where r m 2 = (x -x m -ma)* + (y -y m ¥, (8) 

and the summation with respect to m includes all positive and negative integral values, 
zero being of course excluded. If we neglect terms of the second order in the displacements, 
we find 

dxp ^ k y<>-y m dyo = !L_ S I * ^ %Q-%m , q , 

dt 27ra 2 w m a ' dt 27ra 2 m m 2na 2 ^ m 2 {) 

* " Fhissigkeits- u. Luftwiderstand,' Phys. Zeitschr. xiii. 49 (1911); also Gdtt. Nachr. 1912, 
p. 547. The investigation is only given in outline in these papers; I have supplied various steps. 



226 Vortex Motion [chap, vii 

The first term in the value of dy /dt is to be omitted as being independent of the 
disturbance*. 

Consider now a disturbance of the type 

x m = ae im *, y m = $e im *, (10) 

where <f> may be assumed to lie between and 2rr. If be small this has the character of 
an undulation of wave-length 2?ra/0. We find 

*--* S--* cm 

The arrangement is therefore unstable, the disturbance ultimately increasing as e kt . When 
the wave-length is large compared with a we have 

X = i K 0/a 2 , (13) 

approximately; cf. Art. 234. 

Proceeding next to the case of the symmetrical double row, the positions at time t of 
vortices in the upper and lower rows may be taken to be 

{ma+Ut + x m , \b+y m ), and (na+Ut + x n ', -%b+y n '\ 

respectively, where U denotes the general velocity of advance of the system, and the origin 
is in the plane of symmetry. 

(£> t> $ & 



<p (9 <9 <9 

The component velocities of a vortex in the upper row, e.g. that for which ra = 0, due 
to the remaining vortices of the same row, will be given as before by (9), where the sum 
2m -1 may be omitted. The components due to the vortex n of the lower row will be 

_^ b+yo-Vn < x -x n '-na 

2tt r 2 ' 2tt r 2 

where r n 2 = (x - x n ' -na) 2 + (y - y n ' + b) 2 . 

If we neglect terms of the second order in the disturbance we find, after a little reduction, 

27r(dx \_ y Q -y m s b n 2 a 2 -b 2 



2nab . , s .. .. 

+ !(^w^-*° (1) 



2tt dy _ ^ x -x m n 2 a 2 -b 2 

k dt~ I m 2 a 2 + Z(n 2 a 2 + b 2 ) 2{X ° Xn} 



,2^2TT2V2^0-yn), (15) 



n (n 2 a' + < 

where the summations with respect to n go from — oo to +00, including zero. The terms 
in (14) independent of the disturbance will cancel, since, by (5), 

T-r * 11 1"^ K „ b 

U=— coth — = — 2 



a 2tt n n 2 a 2 + b 2 ' 

* In the summations the vortices are to be taken in pairs equidistant from the origin ; other- 
wise the result would be indeterminate. The investigation may be regarded as applying to the 
central portions of a long, but not infinitely long, row; the term referred to is then negligible. 



156] 



Stability of doable rows 



If we now put 

where 0<$<27r, the equations take the form 
2na 2 da 



k dt 
2ira 2 dp 

k dt 
If we write, for shortness, 

the values of the coefficients are* 



-Ap-Ba'-Cp, 
= -Aa -Ca' + Bff. 



k=b/a, 



2nke in< ^ _ . Jtt0 cosh h (ir — <f>) ir 2 sinh /ccf)\ 
l(ri*+¥j 2 ~ l \ sinh/br sinh 2 /br J ' ' 

{n 2 -k 2 )e in * 7r 2 cosh£<£ 7r<£sinh£(7r-<£) 



(7 = 2 



» (n» + £*)» 



sinh 2 ^7r 



sinh yfc7r 



227 

.(16) 

.(17) 

.(18) 

.(19) 
.(20) 
.(21) 



To deduce the equations relating to the lower row we have merely to reverse the signs 
of < and b, and to interchange accented and unaccented letters. Hence 

27ra 2 da 



= A(3?-Ba + C(3, 



2tt« 2 d& 
k dt 



.(22) 



= Aa' +Ca+Bp. 



The formulae (17) and (22) are the equations of motion of the vortex-system in what may 
be called a normal mode of the disturbance. 



The solutions are of two types. In the first type we have 

a = a', /3=-/3', 



and therefore 



2vra 2 da 
k dt 



= -Ba-(A-C)(3, 



The solution involves exponentials e xt , the values of X being given by 

2tt« 2 



B±J(A 2 -C 2 ). 



In the second type we have 



and therefore 



a=-a', /3 = /3', . 

27ra 2 da „ . . ~. n 
---=Ba-(A + C)P, 

27ra 2 dp 



k dt 

The corresponding values of X are given by 

27ra 2 



= -(A-C)a+Bp. 



\ = B±J(A 2 -C 2 ). 



•(23) 
.(24) 

.(25) 
.(26) 

.(27) 
.(28) 



The summations with respect to n can be derived from the Fourier expansion 
cosh k (ir - <p) 1 Jl 2&cos<£ 2fccos20 [ 
sinhfor ~^\k + ~VTW + ~2 r +k ir + '''\' 



228 Vortex Motion [chap, vii 

Since B is a pure imaginary, whilst A and C are real, it is necessary for stability in 
each case that A 2 should not exceed C 2 for admissible values of <£. Now when $ = 7r we 
find 

,4 + C=4,r 2 tanh 2 |/br, A-C=%7r 2 coth 2 %k7r (29) 

so that A 2 — C 2 is positive. We conclude that both types are unstable. 

Passing to the unsymmetrical case, we denote the positions of the displaced vortices by 

(ma+Vt + x m , $b+y m ), and ((n + %)a+ Vt + x n \ -£&+#»)» 

where V is given by (6). The requisite formulae are obtained by writing n + ^ for n in 
preceding results. 

d> § 3) <S 



The equations (17) and (22) will accordingly apply, provided* 

1 _ fi im<f> ( n i 1A2 _ Z.2 _2 

„ ^(2n + l)ke i( ^ n+ ^^ . ( ir(f> sinh h (tt - 0) 7r 2 sinh^i .__. 

i * = 2 f/„ i 1X2 . 7.2)2 =Z 1 Z3^ EZ + _U2 EZ ) \ dl ) 



{(» + *)* + ' 



{(w + ^) 2 -F}e f(w+ * ) ^ _ 7r 2 cosh^ 7T(frcosh/fc(7r-(fr) 
« {0 + i) 2 + £ 2 } 2 ~ cosh 2 **- cosh/br ( ' 

These values of A, B, C are to be substituted in (25) and (28). As in the former case it is 
necessary for stability that A 2 should not be greater than C 2 . Now when <£ = 7r, (7=0; 
hence A must also vanish, or 

cosh 2 £7r = 2, &7r = -8814, &/« = & = -281 (33) 

The configuration is therefore unstable unless the ratio of the interval between the two 
rows to the distance between consecutive vortices has precisely this value. 

To determine whether the arrangement is stable, under the above condition, for all 
values of <f> from to 2n, let us write for a moment kin - <fi)=x, kir=n, so that 

k 2 A = — \x 2 , k 2 C= \ (fix cosh fix cosh x - fi 2 sinh fi sinh x), (34) 

where x may range between ±fi. Since A is an even and C an odd function of x, it is 
sufficient for comparison of absolute values to suppose x positive. Hence, writing 

y — fi cosh fx cosh x - fi 2 sinh fi x, (35) 

x 

we have to ascertain whether this is positive for 0<x<fi. Since fi= '8814, cosh/x= x /2, 
sinh ii=l t y is positive for x = 0, and it evidently vanishes for x = fi. Again 

-j- = fi cosh /x sinh x + fx 2 sinh fi — — ^ ju 2 sinh/x 1, (36) 

QjCC 00 00 

which is equal to - 1 for #=0, and vanishes for x = fi. Finally, 

d 2 y , , . sinh a? _ „ . , cosh# _ „ . , sinh x . 

-r~2 = fi cosh fi cosh # - ^ sinh p h2^ 2 sinh/* 2/x/sinh/Li — ^— , ...(37) 

CLOG 00 0G Ob 



* The summations with respect to n can be derived from the expansion 
sinh k {tt - <f>) _ 2 jftcos^0 _ &cos|0 
cosh kir 



<f>) _ 2 jk cos %<f> k cos f0 | 



156-157] Stability of double rows 229 

which is easily seen to be positive for all values of x, since (tanh^)/^?<l. Hence as x 
increases from to /x, dy\dx is steadily increasing from - 1 to 0, and is therefore negative. 
Hence y steadily diminishes from its initial positive value to zero, and is therefore positive. 

We conclude that the configuration is definitely stable* except for x= ±/z, when <f> = 
or 27r, in which cases B=0, by (31), and therefore X = 0. Since the disturbed particles 
are then all in the same phase, the reason why the period of disturbance should be infinite 
is easily perceived. 

This unsymmetrical configuration is of special interest because it is exemplified in the 
trail of vortices which is often observed in the wake of a cylindrical body advancing through 
a fluid. This has suggested further researches. 

The effect of lateral rigid boundaries equidistant from the medial line on the stability of 
the configuration has been discussed by Rosenhead f. He finds that as the ratio a/k of the 
interval a between successive vortices in the same row to the distance h between the walls 
increases from zero to *815 the unsymmetrical arrangement is stable only for a definite 
value of 6/a, which decreases continuously from -281 to '256. But when a/h> *815 there is 
stability for a certain range of values of b/a. And when a/h>l'4l9 the configuration is 
stable for all values of b/a. 

The symmetrical configuration, on the other hand, is always unstable. 

157. When, as in the case of a vortex-pair, or a system of vortex-pairs, the 
algebraic sum of the strengths of all the vortices is zero, we may work out a 
theory of the 'impulse,' in two dimensions, analogous to that given in Arts. 119, 
152 for the case of a finite vortex -system. The detailed examination of this 
must be left to the reader. If P, Q denote the components of the impulse 
parallel to x and y, and N its moment about Oz, all reckoned per unit depth 
of the fluid parallel to z, it will be found that 



P = ptfy£dxdy, Q = -pffx^dxdy jy 



N=-%pJS(x* + f)Sdxdy. J 

For instance, in the case of a single vortex-pair, the strengths of the two 
vortices being ± k, and their distance apart c, the impulse is pice, in a line 
bisecting c at right angles. 

The constancy of the impulse gives 

%kx = const., Xfcy — const., j 
%/c (x 2 + y 2 ) = const. ) 

It may also be shewn that the energy of the motion in the present case is 

given by 

T=-\ 9 \\^dxdy = -\plK^ (3) 

When 2k is not zero, the energy and the moment of the impulse are both 
infinite, as may be easily verified in the case of a single rectilinear vortex. 

* This is stated without proof by Karman. 

t Phil. Trans. A, ccviii. 275 (1929). See also Glauert, Proc. Roy. Soc. A, cxx. 34 (1928). 



230 Vortex Motion [chap, vii 

The theory of a system of isolated rectilinear vortices has been put in a very elegant 
form by Kirchhoff *. 

Denoting the positions of the centres of the respective vortices by (# l5 yi), (x 2 , y 2 ), ••• 
and their strengths by k 1} k 2 , • ••> it is evident from Art. 154 that we may write 

Kl dt dy x ' * l dt dccy ' 

dx 1:= _dW fy2_<^ , l (4) 

* 2 dt ~ 83/2 ' 2 dt ~ dx 2 



where w =k~ ^ K i K a^°S r i2i (5) 

Air 

if r 12 denote the distance between the vortices k 1? k 2 . 

Since TF depends only on the relative configuration of the vortices, its value is unaltered 
when #1, # 2 > ••• are increased by the same amount, whence 23^/8^=0, and, in the same 
way, 28TF/8y! = 0. This gives the first two of equations (2), but the proof is not now 
limited to the case of 2k =0. The argument is in fact substantially the same as in 
Art. 154. Again, we obtain from (4) 



/ dx dy\ / dW dW\ 



or if we introduce polar co-ordinates (r u $i\ (r 2 , 2 ), ... for the several vortices, 

2Kr di=-*w (6) 

Since W is unaltered by a rotation of the axes of co-ordinates in their own plane about the 
origin, we have 23 WJ 80=0, whence 

2«r 2 =const., (7) 

which agrees with the third of equations (2), but is free from the restriction there implied. 
An additional integral of (4) is obtained as follows. We have 



s "{ x dt y Tt)-*\ x dx +y ty ) 



~3-»? « 

If every r be increased in the ratio 1 + e, where € is infinitesimal, the increment of W is 
equal to 2er . 8 W/dr. But since the new configuration of the vortex-system is geometrically 
similar to the former one, the mutual distances r 12 are altered in the same ratio 1 + e, and 
therefore, from (5), the increment of W is e/27r.2Ki/c 2 . Hence (8) may be written in the 
form 

^S-c^- (9) 

158. The preceding results are independent of the form of the sections of 
the vortices, so long as the dimensions of these sections are small compared 
with the mutual distances of the vortices themselves. The simplest case is 
when the sections are circular, and it is of interest to inquire whether this 
form is stable. This question has been examined by Kelvin f. 

* Mechanik, c. xx. 

t Sir W. Thomson, "On the Vibrations of a Columnar Vortex," Phil. Mag. (5), x. 155 (1880) 
[Papers, iv. 152]. 



157-158] Stability of a Columnar Vortex 231 

When the disturbance is in two dimensions only, the calculations are very simple. Let 
us suppose, as in Art. 27, that the space within a circle r = a, having the centre as origin, 
is occupied by fluid having a uniform vorticity co, and that this is surrounded by fluid 
moving irrotationally. If the motion be continuous at this circle we have, for r<a, 

*=-i*(« 2 -r 2 ), (i) 

while for r>a, >//•= — \ coa 2 log a/r (2) 

To examine the effect of a slight irrotational disturbance, we assume, for r<a, 

+ = - i a) (a 2 - r 2 ) + A - 8 cos (s0 - <r*),l 

I (3) 

and, for r > a, yjr = - ^ coa 2 log - + A — cos (s0 - at\ 

where s is integral, and a is to be determined. The constant A must have the same 
value in these two expressions, since the radial component of the velocity, —dtyjrdB, must 
be continuous at the boundary of the vortex, for which r=a, approximately. Assuming 
for the equation to this boundary 

r= a + a cos (s6- at), (4) 

we have still to express that the transverse component (dy^/dr) of the velocity is continuous. 
This gives 

A -coa 2 A 

k cor + s — cos (s6 - at) =- 5 — cos (s0 - at). 

a s r a v 

Substituting from (4), and neglecting the square of o, we find 

coo= —2sAja (5) 

So far the work is purely kinematical; the dynamical theorem that the vortex-lines 
move with the fluid shews that the normal velocity of a particle on the boundary must be 
equal to that of the boundary itself. This condition gives 

dr _ d\jr dyj/ dr 
di~ ~ rd6~ dr rd6' 
where r has the value (4), or 

A , i sa fa\ 

aa = s — + icoa.— (o) 

a a 

Eliminating the ratio A /a between (5) and (6) we find 

o-=4(s-l)co (7) 

Hence the disturbance represented by the plane harmonics in (3) consists of a system 
of corrugations travelling round the circumference of the vortex with an angular velocity 

als = (s-l)ls.%<o (8) 

This is the angular velocity in space ; relative to the rotating fluid the angular velocity is 

a/s— |co= -^co/s, (9) 

the direction being opposite to that of the rotation. When 5=2, the disturbed section is 
an ellipse which rotates about its centre with angular velocity £co. 

The three-dimensional oscillations of an isolated columnar vortex-filament have also 
been discussed by Kelvin in the paper cited. The columnar form is found to be stable for 
disturbances of a general character. 

In a recent paper Rosenhead* has examined the stability of the Karman unsymmetrical 
arrangement when the cross-sections are of finite area. The conclusion is that there is 
stability for strictly two-dimensional disturbances, but instability for sinusoidal longitudinal 
deformations, whose wave-length bears less than a certain ratio to the diameter. 

* Proc. Roy. Soc. A, cxxvii. 590 (1930). 



232 Vortex Motion [chap, vti 

159. The particular case of a two dimensional elliptic disturbance can be 
solved without approximation as follows*. 
Let us suppose that the space within the ellipse 

is occupied by liquid having a uniform vorticity a>, whilst the surrounding fluid is moving 
irrotationally. It will appear that the conditions of the problem can all be satisfied if we 
imagine the elliptic boundary to rotate, without change of shape, with a constant angular 
velocity (n, say), to be determined. 

The formula for the external space can be at once written down from Art. 72, 4° ; viz. 

we have 

yj, = %n (a + bfe- 2 * cos 2r)+%a>ab£, (2) 

where £, r\ now denote the elliptic co-ordinates of Art. 71, 3°, and the cyclic constant k has 
been put = 7ruba>. 

The value of \js for the internal space has to satisfy 

H + p=°" ( 3 > 

with the boundary-condition — ^ + -A = —ny> -^ + nx . ^ (4) 

These conditions are both fulfilled by 

+ = ia(Ax* + Btf*), (5) 

provided A + B = l, Aa 2 - Bb 2 = - (a 2 -b 2 ) (6) 

CO 

It remains to express that there is no tangential slipping at the boundary of the 
vortex; i.e. that the values of 8^/3| obtained from (2) and (5) there coincide. Putting 
x=c cosh ^ cost), y = csinh£sinr7, where c=*J(a 2 -b 2 ), differentiating, and equating coeffi- 
cients of cos 2?7, we obtain the additional condition 

-%n(a + b) 2 e- 2 *=Uc 2 (A -B)cosh£sinh$, 

where £ is the parameter of the ellipse (1). This is equivalent to 

A ^ B —i.±g, (7 , 

a) ab y ' 

since, at points of the ellipse, cosh£ = a/c, sinh£ = 6/c. 

Combined with (6) this gives Aa — Bb— -, (8) 

and n= (^bf" < 9) 

When a = 6, this agrees with our former approximate result. 

The component velocities x, y of a particle of the vortex relative to the principal axes 
of the ellipse are given by 

whence we find -=—n\. y—n- (10) 

a 6' b a v ' 

* Kirchhoff, Mechanik, c. xx.; Basset, Hydrodynamics, ii. 41. 



159-159 a] Elliptic Vortex 233 

Integrating, we find x — ka cos {nt + e), y = kbsin (nt+e), (11) 

where k, e are arbitrary constants, so that the relative paths of the particles are ellipses 
similar to the boundary of the vortex, described according to the harmonic law. If %', y' 
be the co-ordinates relative to axes fixed in space, we find 

x = x cos nt - y sin nt = \h (a + b) cos (2nt + e) + ^k(a-b) cos e, 



•(12) 
y' = x sin nt + y cos nt = \ k (a + b) sin (2nt + e) - %k {a - b) sin e. 



The absolute paths are therefore circles described with angular velocity 2n 



159 a. The motion of a solid in a liquid endowed with vorticity is a problem 
of considerable interest, but is unfortunately not very tractable. The only 
exception is when the motion is two-dimensional, and the vorticity uniform. 

Let x , y be the co-ordinates, relative to fixed axes, of a point C of the (cylindrical) 
solid ; let x, y be the co-ordinates of any point of the fluid relative to parallel axes through 
C, and let {u, v) be the velocity relative to C. We have then 



du .. du dv „ 1 dp 

dv .. du dv „ 1 dp 



.(1) 



cf. Arts. 12 (3) and 146 (5). Since 



u-- d -± v- d ± (2) 



and £ is constant, it appears that dujdt and dv/dt are the derivatives with respect to x and 
y, respectively, of a certain function of x, y, t. Denoting this function by - dcfr/dt, we have 

dx\dt)~ dt dy\dt)j dy\dt)~ dt dx\dt)> {) 

which are the conditions that -j- (<p + iyj/) 

should be a function of the complex variable x + iy. This consideration determines dcj)/dt 
when the form of y\r is known t. 

The equations (1) now give 

^ d ^-(x x+y y)-^ + C^, (4) 

where q2 = u 2 + v 2 (5) 

We proceed to apply these results to some cases of motion of a circular cylinder. The 
point C is naturally taken on its axis. 

Let us suppose in the first instance that the undisturbed motion of the fluid consists 
of a uniform rotation co about the origin, so that £=2o>. The stream-function for the 
motion relative to a moving point (x , y ) is then 

^o = i a) {(x + xf + (y +y) 2 } + x y - y x 

= \ (or 2 + cor (x cos 6 + y sin 6) + ^ a> (x 2 +y 2 )+r(x sin 0-y o cos 6), (6) 

* For further researches in this connection see Hill, "On the Motion of Fluid part of which 
is moving rotationally and part irrotationally," Phil. Trans. 1884; Love, «'On the Stability of 
certain Vortex Motions," Proc. Lond. Math. Soc. (1) xxv. 18 (1893). 

f Cf. Proudman, "On the Motion of Solids in a Liquid possessing Vorticity," Proc. Roy. 
Soc. A, xcii. 408 (1916). 



234 Vortex Motion [chap, vii 

where we have introduced polar co-ordinates relative to C. The relative stream-function 
for the disturbed motion will be 

^=^<or 2 + a> (r j (x o cos0+y o sm0) + %a>(xQ Z +yQ 2 ) + (r — - J (ir sin^-^ cos^). 

(7) 

For this satisfies V 1 2 >Jr = 2<o; it makes \jr = const, for r=a; and it agrees with (6) for r=oo . 

Hence 37 = G> \ r ) (%QCOsd+y sm8) + \ r ) (x aind-i/ cos6), (8) 

and therefore 

^r= -a, ( r -\ — J (ir o sin0-y o cos0) + lr-\ — J (#ocos0+j/ o sin0), (9) 

terms independent of r and 6 being omitted. Again we have, for r=a, 

-^ =0, jf- = coa + 2co (x cos 6 +y sin 0) + 2 (i; sin - $ cos 0), 

and therefore 

^ 2 =2cD 2 a(^ o cos0-|-yo sm ^) + 2(aa(Aosin0-^oCos0) + etc., (10) 

where terms are omitted which will contribute nothing to the resultant force on the 
cylinder. Substituting in (4) we find, for r=a, 

V) 

- = a(x o cos0+y o sin 6) - 4a>a (ir sin — $ cos#) - 2o> 2 a (# cos# +y sin 0) + etc. . ..(11) 

The component forces on the cylinder, due to fluid pressure, are therefore* 

/lit 
p cos a d0 = - M' (x Q + 4g># - 2g> 2 # ), 

/2jt 
p sin ad3= —M' (y - 4© x — 2<» 2 y ), 

where M' = npa 2 . Hence if M be the mass per unit length of the cylinder itself, the 
equations of motion are 

H 

fii/ — 4a>x — 2<a 2 y = Y\M',) 

where /u=l + Jf/Jf , and the zero suffixes have been omitted as no longer necessary. If we 
write z=x+iy, these equations are equivalent to 

f£-4:iG>z-2<o 2 z=(X+iY)/M' (14) 

To ascertain the free motion, when X=0, F=0, we assume that z oc e imo,t , and find 

/iwi 2 -4m + 2 = (15) 

If /n<2, i.e. if the mass of the cylinder is less than that of the fluid which it displaces, the 
values of m are real, and the solution has the form 

z^AeWi^ + Be™*"*, (16) 

where m u m 2 are positive. This represents motion in a 'direct 5 epicyclic. As special 
cases circular paths are possible, and are stable. If on the other hand /x>2, the values of 
m are complex, and the solution takes the form 

z=(Ae* t + Be-« t )e i f i \ (17) 

the ultimate path being an equiangular spiral. If /x = 2, we have (m- 1) 2 =0, and 

z=(A+Bt)e i » t (18) 

* Cf. G. I. Taylor, "Motion of Solids in Fluids when the Flow is not Irroiational, " Proc. 
Boy. Soc. A, xciii. 99 (1916). 



.(12) 



fix + 4a>y - 2<o 2 x = X\ M',) 

.(13) 



159 a] Cylinder in Rotating Fluid 235 

Hence, although it is possible as we should expect for a cylinder having the same mean 
density as the fluid to revolve with the latter in a circular path, this motion is unstable. 

If there is a radial force whose direction revolves with the fluid, say 

X+iY=R*~, (19) 

the equation (14) is satisfied, when /* = 2, by 

z = re i <* t , (20) 

provided r=\R\M' (21) 

The cylinder can therefore move, relatively to the rotating fluid, along a radius*, but this 
motion, again, must be classed as unstable t. 

Let us next suppose that the fluid when undisturbed is in laminar motion parallel to 
Ox, with constant vorticity 2<o, the stream -function being 

V'o=a>(yo+.y) 2 =W 2 ( l ~ cos 2 ^) + 2a #<>r sintf + a>y 2 (22) 

In the disturbed motion relative to the cylinder 

^ = ^o>r 2 -|o> (r 2 -\\ cos28 + 2a>y (r- < —\ sm0 + ny o 2 + (r- — J (x sin 6 - y Q cos 6). 

(23) 

Hence -^- = 2(oy (r- — J sin 6+(r J (i? sin 0-y o cos0), (24) 

the terms independent of r and 6 being omitted. We write therefore 

^ = 2coy o (r + ^cos6 + (r+^)(x^co&6+i/o8in0^ (25) 

For r = a we have from (23) 

-=^ = 0, ~= -<oa + 4a>a sin 2 0+4e«>yo sin + 2 (x sin — y cos 0), (26) 

and therefore 

\ q 2 = — 4o) 2 ay sin 6 — 2a>a (x sin 6 - y cos 0) + 1 6o> 2 ay sin 3 

+ 8a>ay (#o sin 2 — i/ Q sin 2 cos 0) + etc., (27) 

those terms only being retained which will contribute to the resultant force on the cylinder. 
Substituting in (4) we find, for r = a, 

P 

—=a(x cos +y sin 0) + 2aax (sin — 4 sin 3 0) + 2o>ay (cos + 4 sin 2 cos 0) 

+ 4o> 2 ay (sin - 4 sin 3 0) + etc (28) 

/•2tt 

Hence % — \ P cos 0ad0= —M' (x + 4g># )> 



/ 



2tt 

jd sin 0ad0 = —M' (i/ — 4a>x — 8o> 2 y ). 



The equations of motion of the cylinder are therefore, omitting the suffixes, 

fix + 4a>y = X/M'j 



(29) 



lii/-4a>x-8 > 2 y=r/M'.! 
We notice that the cylinder can remain at relative rest subject to a force 

T= -8 a > 2 M'y=4 >M'U=2K P U, (31) 

* Cf. Taylor, I.e. 

t Some cases of motion of a sphere in rotating fluid have been studied by Proudman, I.e.; 
S. F. Grace, Proc. Roy. Soc. A, cii. 89 (1922); and Taylor, Proc. Roy. Soc. A, cii. 180 (1922). 
% Cf. Taylor, I.e. 



236 Vortex Motion [chap, vii 

where U{= - 2a>i/) is the velocity of the undisturbed stream at the level of the centre, and 
k ( = 27r« 2 o>) is the circulation immediately round the cylinder. This result may be con- 
trasted with Art. 69 (6). 

It is easily found from (30) that, if /i<2, the path when there are no extraneous 
forces is a trochoid whose general direction of advance is parallel to the stream. 

160. It was pointed out in Art. 80 that the motion of an incompressible 
fluid in a curved stratum of small and uniform thickness is completely denned 
by a stream-function yjr, so that any kinematical problem of this kind may be 
transformed by projection into one relating to a plane stratum. If, further, 
the projection be 'orthomorphic,' the kinetic energy of corresponding portions 
of liquid, and the circulations in corresponding circuits, are the same in the 
two motions. The latter statement shews that vortices transform into vortices 
of equal strengths. It follows at once from Art. 145 that in the case of a closed 
simply-connected surface the algebraic sum of the strengths of all the vortices 
present is zero. 

We may apply this to motion in a spherical stratum. The simplest case is that of a 
pair of isolated vortices situated at antipodal points ; the stream-lines are then parallel 
small circles, the velocity varying inversely as the radius of the circle. For a vortex-pair 
situate at any two points A, B, the stream-lines are coaxal circles as in Art. 80. It is 
easily found by the method of stereographic projection that the velocity at any point P is 
the resultant of two velocities k fair a . cot ^0 1 and ufa-ira . cot jj0 2 > perpendicular respectively 
to the great-circle arcs AP, BP, where U 2 dbriote the lengths of these arcs, a the radius 
of the sphere, and ±< the strengths of the vortices. The centre* (see Art. 154) of either 
vortex moves perpendicular to AB with a velocity k/27t« . cot \AB. The two vortices 
therefore describe parallel and equal small circles, remaining at a constant distance from 
each other. 

Circular Vortices. 

161. Let us next take the case where all the vortices present in the liquid 
(supposed unlimited as before) are circular, having the axis of x as a common 
axis. Let to- denote the distance of any point P from this axis, v the velocity 
in the direction of ot, and a> the resultant vorticity at P. It is evident that 
u, v, co are functions of x, zr only. 

Under these circumstances there exists a stream-function yjr, defined as in 
Art. 94, viz. we have 

m= _i^ ¥ _i£ 

tar c-gt ts ox 

whence a> = -- — =.- ( *+* JL\ (2) 

ox ost nr\oar o-sr m otz ) 

It is easily seen from the expressions (7) of Art. 148 that the vector 
(F, G, H) will under the present conditions be everywhere perpendicular to 

* To prevent possible misconception it may be remarked that the centres of corresponding 
vortices are not necessarily corresponding points. The paths of these centres are therefore not 
in general projective. 



i59a-i6i] Circular Vortices 237 

the axis of x and the radius -&. If we denote its magnitude by S, the flux 
through the circle (x, m) will be 2ir^S, whence 

^ = -*tS (3) 

To find the value of i/r at (x, ©■) due to a single vortex-filament of circulation 

tc, whose co-ordinates are x', w', we note that the element which makes an 

angle 6 with the direction of S may be denoted by vt'hd, and therefore by 

Art. 149 (1) 

ic&w' f 27r cos 6 ■ 

t— rf ^— 5tJ — m, w 

where r = {(x- x'f + w 2 + w' 2 - 2*nsr' cos 0}i (5) 

If we denote by r 1} r 2 the least and greatest distances, respectively, of the 
point P from the vortex, viz. 

r 2 = (x- x') 2 + O - m') 2 , r 2 2 = (x - x'f + (# + OT ') 2 , (6) 

we have r 2 = j\ 2 cos 2 J + ?' 2 2 sin 2 J #, 4<*tot-' cos 6 — r ± 2 + r 2 2 — 2r 2 , (7) 

and therefore 



i K 



(n 2 + r 2 2 ) J 



V(ri 2 cos 2 ^ + r 2 2 sin 2 ^) 



2 f V (n 2 cos 2 i (9 + r 2 2 sin 2 § (9) dd 

Jo 



(8) 



The integrals are of the types met with in the theory of the ' arithmetico- 
geometrical mean.'* In the ordinary, less symmetrical, notation of 'complete' 
elliptic integrals we have 

t = -^(-')*{g-^ lW -|^ W }, (9) 



provided y-l-gg ^- ^ ^ , x2 (10) 



4gg' 
r 2 2 (0 - x') 2 + (tsr + t*') 2 
The value of -^ at any assigned point can therefore be computed with the 
help of Legendre's tables. 

A neater expression may be obtained by means of 'Landen's trans- 
formation,' f viz. 

f = -£(r l + r 2 ){F 1 (X)-E 1 (X)}, (11) 

provided x^HZl* (12) 

* ri + ri v 

The forms of the stream-lines corresponding to equidistant values of yjs are shewn on 
the next page. They are traced by a method devised by Maxwell, to whom the formula (11) 
is also due J. 

* See Cayley, Elliptic Functions, Cambridge, 1876, c. xiii. 
t See Cayley, I.e. 

J Electricity and Magnetism, Arts. 704, 705. See also Minchin, Phil. Mag. (5), xxxv. (1893); 
Nagaoka, Phil. Mag. (6), vi. (1903). 



238 



Vortex Motion 



CHAP. VII 



Expressions for the velocity-potential and the stream-function can also be obtained in 
the fcrm of definite integrals involving Bessel's Functions. 




X'- 




Thus, supposing the vortex to occupy the position of the circle x = 0, tzr=a, it is evident 
that the portions of the positive side of the plane x = which lie within and without this- 



161-162] Stream-lines of a Vortex Ring 239 

circle constitute two distinct equipotential surfaces. Hence, assuming that we have $ = %< 
for # = 0, w<a, and <£=0 for #=0, w>a, we obtain from Art. 102 (2) 

= i K a f ^ e- kx J Q {kw)J l {ka)dk, (13) 

and therefore, in accordance with Art. 100 (5), 

ty=-% K aw J e-te^^Jiikcfidk (14) 

These formulae relate of course to the region x >0*. 

It was shewn in Art. 150 that the value of <p is that due to a system of double sources 
distributed with uniform density k over the interior of the circle. The values of cf> and -v//- 
for a uniform distribution of simple sources over the same area have been given in Art. 102 
(11). The above formulae (13) and (14) can thence be derived by differentiating with 
respect to x, and adjusting the constant factor f. 

162. The energy of any system of circular vortices having the axis of x as 
a common axis, is 

T= irp \(u 2 + v 2 ) ■urdxd'n = 7rp I j(v ~ — u~-j dxdts 

= — irp I jyfrcodxd'OT = — 7rpX«^, (1) 

by a partial integration, the integrated terms vanishing at the limits. We have 
here used k to denote the strength coSxSzz of an elementary vortex-filament. 

Again the formula (7) of Art. 153 becomes^ 

T = 27T/o//(OTtt — xv) mwdxdy— 2TrpX/cur {tzu —xv) (2) 

The impulse of the system obviously reduces to a force along Ox. By 
Art. 152 (6), 

P = J p fj \y% ' — zrj) dxdydz = irp Jj^codxd'ST = rrpX/m* (3) 

If we introduce the symbols x , w denned by the equations 

2/csr 2 # 9 2/eor 2 
^o=^ 2, W = ~^—, (4) 

these determine a circle whose position evidently depends on the strengths 
and the configuration of the vortices, and not on the position of the origin on 
the axis of symmetry. It may be called the 'circular axis' of the whole system 
of vortex-rings. 

* The formula for \j/ occurs in Basset, Hydrodynamics, ii. 93. See also Nagaoka, I.e. 

t Other expressions for <f> and x}/ can be obtained in terms of zonal spherical harmonics. 
Thus the value of <p is given in Thomson and Tait, Art. 546 ; and that of \f/ can be deduced by 
the formulae (11), (12) of Art. 95 ante. The elliptic-integral forms are however the most useful 
for purposes of interpretation. 

{ At any point in the plane 2 = we have y = w, £ = 0, y = 0, f=£w, v = v; the rest follows by 
symmetry. 



240 Vortex Motion [chap, vii 

Since k is constant for each vortex, the constancy of the impulse shews, 
by (3) and (4), that the circular axis remains constant in radius. To find its 
motion parallel to x, we have, from (4), 

Ik . ctq 2 . -jj — It/cn 2 -=r + 2%/cgfx -77 = Xkhf (vtu + 2xv) (5) 

With the help of (2) this can be put in the form 

2/C.OTo 2 . -37 = 2^" + 32* (x - COq) VTV, (6) 

where the added term vanishes, since ^kutv = on account of the constancy 
of the mean radius (txr ). 

163. Let us now consider, in particular, the case of an isolated vortex-ring 
the dimensions of whose cross-section are small compared with the radius (-5t ). 
It has been shown that 

♦-^//K^)-*^)}^^^^ « 

where r 1} r 2 are defined by Art. 161 (6). For points (x, m) in or near the 
substance of the vortex, the ratio r\\r% is small, and the modulus (X) of the 
elliptic integrals is accordingly nearly equal to unity. We then have 

^ 1 (X) = |logi|, &W-1, (2) 

approximately*, where X' denotes the complementary modulus, viz. 

x ^ 1 - x, -5S3?' (3 > 

or X' 2 = 5ri/r 2 , nearly. 

Hence at points within the substance of the vortex the value of yjr is of 
the order /csy log (wo/e), where e is a small linear magnitude comparable with 
the dimensions of the section. The velocities at such points, depending 
(Art. 94) on the differential coefficients of yfr, will be of the order x/e. 

We can now estimate the magnitude of the velocity dx /dt of translation 
of the vortex-ring. By Art. 162 (1), T is of the order ptc 2 Gr \og (sr /e), and v is, 
as we have seen, of the order /e/e; whilst x — x is of course of the order e. 
Hence the second term on the right-hand side of the formula (6) of the 
preceding Art. is, in the present case, small compared with the first, and the 
velocity of translation of the ring is of the order k/vfq . log (w /e), and approxi- 
mately constant. 

An isolated vortex-ring moves then, without sensible change of size, parallel 
to its rectilinear axis with nearly constant velocity. This velocity is small 
compared with that of the fluid in the immediate neighbourhood of the circular 
axis, but may be greater or less than J/c/oto, the velocity of the fluid at the 
centre of the ring, with which it agrees in direction. 

* See Cayley, Elliptic Functions, Arts. 72, 77; and Maxwell, I.e. 



162-163] Speed of a Vortex-Ring 241 

For the case of a circular section more definite results can be obtained as follows. If 
we neglect the variations of w and a> over the section, the formulae (1) and (2) give 

*=-£-.//(i°g 8 f°-*)<w 

or, if we introduce polar co-ordinates (s, %) in the plane of the section, 

♦--E-tf/r^-")'**' (4) 

where a is the radius of the section. Now 

I ' log r x dx=f * log {s 2 + s' 2 - 2ss' cos ( x - xjfi d Xi 

and this definite integral is known to be equal to 2tt logs', or 2?r log s, according as s'^s. 
Hence, for points within the section, 

*- - o. CTo f (log §p - 2) s'M - vwJ^ (log ^r»-a) *' *' 

= -Ja^a* {log^-f-^} (5) 

The only variable part of this is the term ^a>w s 2 ; this shews that to our order of approxi- 
mation the stream-lines within the section are concentric circles, the velocity at a distance 
s from the centre being ^a>s. 

Substituting in Art. 162 (1) we find 



277-p 
The last term in Art. 162 (6) is equivalent to 



h=-H:!>^-£^-*} (e) 



f ot co2k (x-Xq) 2 . 



In our present notation, where k denotes the strength of the whole vortex, this is equal to 
f K 2 ar /7r. Hence the formula for the velocity of translation of the vortex becomes* 



dx < L 8tsr ] 



•(7) 



The vortex-ring carries with it a certain body of irrotationally moving fluid in its 
career; cf. Art. 155, 2°. According to the formula (7) the velocity of translation of the 
vortex will be equal to the velocity of the fluid at its centre when zzr /a = 86, about. The 
accompanying mass will be ring-shaped or not, according as or /a exceeds or falls short of 
this critical value. 

The ratio of the fluid velocity at the periphery of the vortex to the velocity at the centre 
of the ring is 2a>ai37 //c, or -as^na. For a = T JoZ<r , this is equal to 32, about. 

The conditions under which a vortex-ring of finite section and uniform 
vorticity can travel unchanged have been investigated by Lichtenstein f . The 
shape of the section, when small, is found to be approximately elliptic, with 
the minor axis in the direction of translation. He has also discussed the 
analogous question relating to a vortex-pair (Art. 155). 

* This result was given without proof by Sir W. Thomson in an appendix to a translation of 
Helmholtz' paper, Phil. Mag. (4), xxxiii. 511 (1867) [Papers, iv. 67]. It was verified by Hicks, 
Phil. Trans. A, clxxvi. 756 (1885); see also Gray, "Notes on Hydrodynamics," Phil. Mag. (6), 
xxviii. 13 (1914). 

t Math. Zeitsch. xxiii. 89, 310 (1925). See also his Grundlagen der Hydrodynamik, Berlin, 
1829. 



242 Vortex Motion [chap, vn 

164. If we have any number of circular vortex-rings, coaxal or not, the 
motion of any one of these may be conceived as made up of two parts, one 
due to the ring itself, the other due to the influence of the remaining rings. 
The preceding considerations shew that the second part is insignificant com- 
pared with the first, except when two or more rings approach within a very 
small distance of one another. Hence each ring will move, without sensible 
change of shape or size, with nearly uniform velocity in the direction of its 
rectilinear axis, until it passes within a short distance of a second ring. 

A general notion of the result of the encounter of two rings may, in par- 
ticular cases, be gathered from the result given in Art. 149 (3). Thus, let us 
suppose that we have two circular vortices having the same rectilinear axis. 
If the sense of the rotation be the same for both, the two rings will advance, 
on the whole, in the same direction. One effect of their mutual influence will 
be to increase the radius of the one in front, and to contract the radius of the 
one in the rear. If the radius of the one in front becomes larger than that of 
the one in the rear, the motion of the former ring will be retarded, and that 
of the latter accelerated. Hence if the conditions as to relative size and 
strength of the two rings be favourable, it may happen that the second ring 
will overtake and pass through the first. The parts played by the two rings 
will then be reversed ; the one which is now in the rear will in turn overtake 
and pass through the other, and so on, the rings alternately passing one through 
the other*. 

If the rotations be opposite, and such that the rings approach one another, 
the mutual influence will be to enlarge the radius of each. If the two rings 
be moreover equal in size and strength, the velocity of approach will continually 
diminish. In this case the motion at all points of the plane which is parallel 
to the two rings, and half-way between them, is tangential to this plane. We 
may therefore, if we please, regard the plane as a fixed boundary to the fluid 
on either side, and so obtain the case of a single vortex-ring moving directly 
towards a fixed rigid wall. 

The foregoing remarks are taken from Helm hoi tz' paper. He adds, in 
conclusion, that the mutual influence of vortex-rings may easily be studied 
experimentally in the case of the (roughly) semicircular rings produced by 
drawing rapidly the point of a spoon for a short space through the surface of 
a liquid, the spots where the vortex-filaments meet the surface being marked 
by dimples. (Cf. Art. 27.) The method of experimental illustration by means 
of smoke-rings | is too well-known to need description here. A beautiful 

* Cf. Hicks, "On the Mutual Threading of Vortex Kings," Proc. Rqy. Soc. A, cii. Ill (1922). 
The corresponding case in two dimensions was worked out and illustrated graphically by Grobli, 
I.e. ante p. 224; see also Love, "On the Motion of Paired Vortices with a Common Axis," Proc. 
Lond. Math. Soc. xxv. 185 (1894), and Hicks, I.e. 

f Reusch, " Ueber Ringbildung der Flussigkeiten," Pogg. Ann. ex. (1860); Tait, Recen 
Advances in Physical Science, London, 1876, c. xii. 



164-165] Mutual Influence of Vortex-Rings 243 

variation of the experiment consists in forming the rings in water, the sub- 
stance of the vortices being coloured*. 

The motion of a vortex-ring in a fluid limited (whether internally or externally) by a 
fixed spherical surface, in the case where the rectilinear axis of the ring passes through 
the centre of the sphere, has been investigated by Lewis f, by the method of 'images.' 
The following simplified proof is due to Larmor %. The vortex-ring is equivalent (Art. 150) 
to a spherical sheet of double-sources of uniform density, concentric with the fixed sphere. 
The ' image ' of this sheet will, by Art. 96, be another uniform concentric double-sheet, 
which is, again, equivalent to a vortex-ring coaxal with the first. It easily follows from 
the Art. last cited that the strengths (k, k') and the radii (sr, g/) of the vortex-ring and 
its image are connected by the relation 

kw% + k'w'% = (1) 

The argument obviously applies to the case of a re-entrant vortex of any form, provided 
it lie on a sphere concentric with the boundary. 

The interest attaching to Karman's stable configuration of a system of 
line- vortices of small section (Art. 156) has led to the discussion of analogous 
arrangements in three dimensions. 

Considering, in the first instance, a procession of equal vortex-rings of 
infinitesimal section, spaced at equal intervals with a common axis, Levi and 
Forsdyke§ find that the arrangement is unstable for a type of disturbance in 
which the radii and the intervals vary simultaneously, the rings remaining 
accurately plane and circular. On the other hand, provided the ratio of the 
interval between successive rings to the common radius exceeds 1*20, periodic 
vibrations about the circular form are possible, of types discussed by 
J. J. Thomson and Dyson in the case of an isolated ring||. 

They examine next the case of a helical vortex 1F. If undisturbed this will 
have a certain angular velocity about its axis, and a certain velocity of advance. 
They find that there is stability if, and only if, the pitch of the helix 
exceeds 03. 

The Conditions for Steady Motion. 
165. In steady motion, i.e. when 

^ = ^ = ^ = 

dt ' dt ' dt u ' 

the equations (2) of Art. 6 may be written 

du dv t dw , . dfl ldp .... 

w d-x +v d- x +w te-w- wr >^-te-- P £ (1) 

* Keynolds, "On the Kesistance encountered by Vortex Kings &c," Brit. Ass. Rep. 1876; 
Nature, xiv. 477. 

f " On the Images of Vortices in a Spherical Vessel," Quart. Journ. Math. xvi. 338 (1879). 

X "Electro-magnetic and other Images in Spheres and Planes," Quart. Journ. Math, xxiii. 
94 (1889). 

§ Proc. Roy. Soc. A, cxiv. 594; A, cxvi. 352 (1927). 

|| For references see p. 246. H Proc. Roy. Soc. A, cxx. 670 (1928). 



244 Vortex Motion [chap, vii 

Hence, if as in Art. 146 we put 

x' = /f + i2 2 +n. (2) 

we have ^—v^ — wr^ ^r- = wi; — u£, -~—ur]-v^ (3) 

It follows that u-% +v£- + w-£- = 0, 

ox oy oz 

so that each of the surfaces tf = const, contains both stream-lines and vortex- 
lines. If further 8n denote an element of the normal at any point of such a 

surface, we have 

dy' 

^ = qw sin/3, (4) 

where q is the current velocity, co the vorticity, and /3 the angle between the 
stream-line and the vortex-line at that point. 

Hence the conditions that a given state of motion of a fluid may be a 
possible state of steady motion are as follows. It must be possible to draw in 
the fluid an infinite system of surfaces each of which is covered by a network 
of stream -lines and vortex-lines, and the product qw sin @8n must be constant 
over each such surface, 8n denoting the length of the normal drawn to a con- 
secutive surface of the system *. 

These conditions may also be deduced from the considerations that the 
stream-lines are, in steady motion, the actual paths of the particles, that the 
product of the angular velocity into the cross-section is the same at all points 
of a vortex, and that this product is, for the same vortex, constant with regard 
to the time. 

The theorem that the function %', defined by (2), is constant over each 
surface of the above kind is an extension of that of Art. 21, where it was shewn 
that x is constant along a stream-line. 

The above conditions are satisfied identically in all cases of irrotational 
motion, provided of course the boundary-conditions be such as are consistent 
with the steady motion. 

In the motion of a liquid in two dimensions (xy) the product q8n is con- 
stant along a stream-line ; the conditions in question then reduce to this, that 
the vorticity £ must be constant along each stream-line, or, by Art. 59 (5), 

3+^=/w. < 5 > 

where /(^) is an arbitrary function of yfr f. 

* See a paper "On the Conditions for Steady Motion of a Fluid," Proc. Lond. Math. Soc. (1) 
ix. 91 (1878). 

t Cf. Lagrange, Nouv. Mem. de VAcad. de Berlin, 1781 [Oeuvres, iv. 720]; and Stokes, "On 
the Steady Motion of Incompressible Fluids," Gamb. Trans, vii. (1842) [Papers, i. 15]. 



165] Conditions for Steady Motion 245 

This condition is satisfied in all cases of motion in concentric circles about the origin. 
Another obvious solution of (5) is 

+ = \{Ax* + 2Bxy + Cy*\ (6) 

in which case the st eam-lines are similar and coaxal conies. The angular velocity at any 
point is -| (A + C), an 1 is therefore uniform. 

Again, if we put j (\jr) = - k 2 \js, where k is a constant, and transform to polar co-ordinates 
r, 0, we get 

which is satisfied (Art. 101) by ty = CJ s {kr) C08 \ sd (8) 

This gives various solutions consistent with a fixed circular boundary of radius w, the 
admissible values of k being determined by 

J s (ka)=0 (9) 

Suppose, for example, that in an unlimited mass of fluid the stream-function is 

+ = CJi(kr)sm6, (10) 

within the circle r=a, whilst outside this circle we have 



f=u(r~\smd. 



.(11) 



These two values of \js agree for r=a, provided J x (ka)=0. Moreover, the tangential velocity 
at this circle will be continuous, provided the two values of d^jrjdr are equal, i.e. if 

W(ba) kJ (ka) y } 

If we now impress on everything a velocity U parallel to Ox, we get a species of cylindrical 
vortex travelling with velocity U through a liquid which is at rest at infinity. The smallest 
of the possible values of k is given by kaf-n- = 1'2197 ; the relative stream-lines inside the 
vortex are then given by the lower diagram on p. 288, provided the dotted circle be taken 
as the boundary (r=a). It is easily proved, by Art. 157 (I), that the 'impulse' of the vortex 
is represented by 2irpa 2 U. 

In the case of motion symmetrical about an axis (%), we have q . 27rtsrSn 
constant along a stream-line, -cr denoting as in Art. 94 the distance of any 
point from the axis of symmetry. The condition for steady motion then is 
that the ratio a/vr must be constant along any stream-line. Hence, if t/t be 
the stream-function, we must have, by Art. 161 (2), 

a^ + a^-- a — -VW> (is) 

where /(-v/r) denotes an arbitrary function of yfr*. 

An interesting example is furnished by Hill's ' Spherical Vortex f.' If we assume 

^=4^37 2 (a 2 -r 2 ) (14) 

where r 2 =# 2 + zsr 2 , for all points within the sphere r=a, the formula (2) of Art. 161 makes 

0)= —^A'STy 

so that the condition of steady motion is satisfied. Again it is evident, on reference to 
Arts. 96, 97, that the irrotational flow of a stream with the general velocity — U parallel to 
the axis, past a fixed spherical surface r=a, is given by 



+-lw(i-£) ( 15 > 



* This result is due to Stokes, I.e. 

f "On a Spherical Vortex," Phil. Trans. A, clxxxv. (1894). 



246 



Yortex Motion 



[chap, vii 



The two values of y\r agree when r=a ; this makes the normal velocity zero on both sides. 
In order that the tangential velocity may be continuous, the values of dyjf/dr must also 
agree. Remembering that or=r sin 0, this gives A = — § U/a 2 , and therefore 

a> = ^-U^ja 2 (16) 

The sum of the strengths of the vortex-filaments composing the spherical vortex is 5 Ua. 

The figure shews the stream-lines, both inside and outside the vortex ; they are drawn, 
as usual, for equidistant values of x//\ 




If we impress on everything a velocity U parallel to x, we get a spherical vortex 
advancing with constant velocity U through a liquid which is at rest at infinity. 

By the formulae of Art. 162, we readily find that the square of the ' mean-radius ' of the 
vortex is fa 2 , the 'impulse' 2irpa 3 U, and the energy is ^-Trpa^U 2 . 

As explained in Art. 146, it is quite unnecessary to calculate formulae for the pressure, 
in order to assure ourselves that this is continuous at the surface of the vortex. The con- 
tinuity of the pressure is already secured by the continuity of the velocity, and the constancy 
of the circulation in any moving circuit. 

166. As already stated, the theory of vortex motion was originated by 
Helmholtz in 1858. It acquired additional interest when, in 1867, Kelvin 
suggested* the theory of vortex atoms. As a physical theory, this has long 
been abandoned, but it gave rise to a great number of interesting investi- 
gations, to which some reference should be made. We may mention the 
investigations as to the stability and the periods of vibration of rectilinear f 
and annular J vortices ; the similar investigations relating to hollow vortices 
(where the rotationally moving core is replaced by a vacuum§); and the cal- 
culations of the forms of boundary of a hollow vortex which are consistent 
with steady motion ||. A summary of some of the leading results has been 
given by Love IT. 

* I.e. ante p. 222. 

+ Sir W. Thomson, I.e. ante p. 230. 

% J. J. Thomson, I.e. ante p. 216 ; Dyson, Phil. Trans. A, clxxxiv. 1041 (1893). 

§ Sir W. Thomson, I.e. ; Hicks, "On the Steady Motion and the Small Vibrations of a 
Hollow Vortex," Phil. Trans. 1884; Pocklington, "The Complete System of the Periods of a 
Hollow Vortex King," Phil. Trans. A, clxxxvi. 603 (1895); Carslaw, "The Fluted Vibrations of 
a Circular Vortex-King with a Hollow Core," Proc. Lond. Math. Soc. (1) xxviii. 97 (1896). 

|| Hicks, I.e.; Pocklington, "Hollow Straight Vortices," Camb. Proc. viii. 178 (1894). 

IF I.e. ante p. 192. 



165-166 a] Bjerknes' Theorem 247 

166 a. The dynamical theorems of the present chapter all depend on the 
constancy of the circulation in a moving circuit. It is postulated (Art. 146) 
that the extraneous forces if any are conservative, and also that the fluid is 
either homogeneous and incompressible, or subject to a definite relation 
between the pressure and the density. 

There are of course many natural conditions, especially in Meteorology, in 
which this latter assumption does not hold. If we proceed as in Art. 33 
without making this assumption we find, for the rate of change of the circu- 
lation in a moving circuit, 

^J{udx + vdy + wdz) = -j8\^dx + ^dy+^dzj t (1) 

where s (= 1/p) is the reciprocal of the density, or the 'bulkiness', of the fluid. 
The line-integral on the right hand may be converted into a surface-integral 
over any area bounded by the circuit, by Stokes' theorem ; thus 

jyl(urda> + vdy+wdz)= \\{IP + mQ + nlt)dS, (2) 

where *-|M C = fM fi-?4*4 (3) 

d(y,z) d(z t a>) o{x,y) 

Now consider the vector whose components are P, Q y R. It is solenoidal, in 
virtue of the relation 

dP dQ dB 

dx + d y + a* ' w 

and its direction is given by the intersections of the surfaces p = const., s = const. 
If we imagine a series of surfaces of equal pressure to be drawn for equal 
infinitesimal intervals 8p } and likewise a series of surfaces of equal bulkiness 
for equal infinitesimal intervals 8s, these will divide the field into a system of 
tubes whose cross-sections are infinitesimal parallelograms. It is easy to shew 
that if 8% is the area of one of these parallelograms 

*J(P*+Q 2 + R 2 )82 = 8p8s (5) 

Hence the product of the vector (P, Q, R,) into the cross-section is not only 
uniform along any tube, but is the same for all the tubes. The equation (2) 
then shews that the rate of change of the circulation round a moving circuit 
is proportional to the number of the aforesaid tubes which it embraces *. 

* V. Bjerknes, Vid.-Selsk. Skrifter, Kristiania, 1918. An independent proof is attributed to 
Silberstein (1896). Another theorem of a less simple character is given by Bjerknes, relating to 
the circulation of momentum 

jp (udx + vdy + wdz). 

Some applications of the theorems to meteorological and other phenomena are explained in 
Stockholm, Ah. Handl. xxxi. (1898). 



248 Vortex Motion [chap, vii 

Clebsch's Transformation. 

167. Another matter of some interest, which can however only be briefly 
touched upon, is Clebsch's transformation of the hydrodynamical equations*. 

It is easily seen that the component velocities at any one instant can be expressed in 
the forms 

— B+>£. — ! +x !> —£+4 t» 

where <£, X, fi are functions of x, y, z, provided the component rotations can be put in the 
forms 

dQ^jx) 3(X,/i) d(\ it i) 

t-d&zy *""a («,*)• i d{*,y) {) 

Now if the differential equations of the vortex-lines, viz. 

dx _dy _ dz . . 

7~t~7' w 

be supposed integrated in the form 

a = const., /3 = const., , (4) 

where a, /3 are functions of x, y, z, we must have 

^-^(y,*)' '-^fc*)* f "^a(*,y)' (& 

where P is some function of #, y, zf . Substituting these expressions in the identity 

d A+h + d A =0 

dx + dy + dz ' 

we find W lJ ^=°> W 

which shews that P is of the form /(a, /3). If X, /x be any two functions of a, /3, we have 

8.0,11) ' 3(X, M ) 3(«,g) 

3(y >2 )-3(« ( » X 8(y )Z )' fflC -'' SC -' 

and the equations (5) will therefore reduce to the form (2), provided X, \i be chosen so that 

S8f$-/hA w 

which can obviously be satisfied in an infinity of ways. 

It is evident from (2) that the intersections of the surfaces X = const., fi= const, are the 
vortex-lines. This suggests that the functions X, \i which occur in (1) may be supposed to 
vary continuously with t in such a way that the surfaces in question move with the fluid % . 
Various analytical proofs of the possibility of this have been given ; the simplest, perhaps, 
is by means of the equations (2) of Art. 15, which give (as in Art. 17) 

udx + vdy + wdz=u da + v db + w dc- dx (8) 

It has been proved that we may assume, initially, 

u da+v db + w dc= —d<t>o + \d[x (9) 

Hence, considering space-variations at time t, we shall have 

udx-\-vdy + wdz= — c?0 + Xcfyi, (10) 

* "Ueber eine allgemeine Transformation d. hydrodynamischen Gleichungen," Crelle, liv. 
(1857) and lvi. (1859). See also Hill, Quart. Journ. Math. xvii. (1881), and Camb. Trans, xiv. 
(1883). 

f Cf. Forsyth, Differential Equations, Art. 174. 

% It must not be overlooked that on account of the insufficient determinacy of X, n these 
functions may vary continuously with t without relating always to the same particles of fluid, 



167] HUTs Spherical Vortex 249 

where (p=cp + x, and X, a have the same values as in (9), but are now expressed in terms 
of x, y, z, t. Since, in the ' Lagrangian ' method, the independent space- variables relate to 
the individual particles, this proves the theorem. 

On this understanding the equations of motion can be integrated, provided the extraneous 
forces have a potential, and that p is a function of p only. We have 

du o y . c* 9« , / 3X , 3X , d\\ da / 3u, da da\ d\ 

~dx\ dt^ dt)* Dt dx Dt dx'' k ; 

and therefore, on the present assumption that D\jDt = 0, DajDt = 0, 



l 



* +Wt „.|-x| (U> 



by Art. 146 (5), (6). An arbitrary function of t is here supposed incorporated in d<p/dt. 
If the above condition be not imposed on X, a, we have, writing 

*-/4 + fc. + 0-j£ + xS (13) 

Dtdx Dtdx~ dx' Dt dy Dt dy~ dy ' Dt dz Dtdz~ dz ' '"^ ' 

Hence l ( ^H } =0, (15) 

d(x,y,z) 

shewing that H is of the form/(X, a, t) ; and 

Dt~ d*> Dt~d\ ""^ } 

* The author is informed that these equations were given in a Fellowship dissertation 
(Dublin) by Mr T. Stuart (1900). 



CHAPTER VIII 

TIDAL WAVES 

168. One of the most interesting and successful applications of hydro- 
dynamical theory is to the small oscillations, under gravity, of a liquid having 
a free surface. In certain cases, which are somewhat special as regards the 
theory, but very important from a practical point of view, these oscillations 
may combine to form progressive waves travelling with (to a first approxi- 
mation) no change of form over the surface. 

The term 'tidal,' as applied to waves, has been used in various senses, but 
it seems most natural to confine it to gravitational oscillations possessing the 
characteristic feature of the oceanic tides produced by the action of the sun 
and moon. We have therefore ventured to place it at the head of this Chapter, 
as descriptive of waves in which the motion of the fluid is mainly horizontal, 
and therefore (as will appear) sensibly the same for all particles in a vertical 
line. This latter circumstance greatly simplifies the theory. 

It will be convenient to recapitulate, in the first place, some points in the 
general theory of small oscillations which will receive constant exemplification 
in the investigations which follow*. The theory has reference in the first 
instance to a system of finite freedom, but the results, when properly inter- 
preted, hold good without this restriction f. 

Let q ly q%, ... q n be n generalized co-ordinates serving to specify the con- 
figuration of a dynamical system, and let them be so chosen as to vanish in 
the configuration of equilibrium. The kinetic energy T will be a homogeneous 
quadratic function of the generalized velocities q lt q 2 , ... q nt say 

2T=a U 5i 2 + a 2 2?2 2 +... + 2a 12 g 1 g 2 + ..., (1) 

where the coefficients are in general functions of the co-ordinates q 1} q 2 , ... q n , 
but may in the application to small motions be supposed constant, and to have 
the values corresponding to q 1} q 2 , ... q n — 0. Again, if (as we shall suppose) 
the system is 'conservative,' the potential energy Fof a small displacement is 
a homogeneous quadratic function of the component displacement q lt q 2} ... q n> 
with (on the same understanding) constant coefficients, say 

2V = c u gi 2 4- c 22 q 2 2 + ... + 2ci2?ig 2 + (2) 

* For a fuller account of the general theory see Thomson and Tait, Arts. 337, ...; Kayleigh, 
Theory of Sound, c. iv.; Bouth, Elementary Rigid Dynamics (6th ed.), London, 1897, c. ix.; 
Whittaker, Analytical Dynamics, c. vii. ; Lamb, Higher Mechanics, 2nd ed., Cambridge, 1929. 

t The steps by which a rigorous transition can be made to the case of infinite freedom have 
been investigated by Hilbert, Gott. Nachr. 1904. 



168] Free Oscillations 251 

By a real* linear transformation of the co-ordinates q lf q 2 , ... q n it is 
possible to reduce T and V simultaneously to sums of squares; the new 
variables thus introduced are called the 'normal co-ordinates' of the system. 
In terms of these we have 

2T=a 1 j 1 2 + a 2 ^+... + a„^ (3) 

2F= c iqi 2 + c 2 q 2 * + ... + c n q n 2 ..(4) 

The coefficients a 1} a 2 , ... a n are called the 'principal coefficients of inertia'; 
they are necessarily positive. The coefficients Ci, c 2 , ... c n may be called the 
'principal coefficients of stability'; they are all positive when the undisturbed 
configuration is stable. 

When given extraneous forces act on the system, the work done by these 
during an arbitrary infinitesimal displacement Aq 1} Aq 2 ,... Ag n may be ex- 
pressed in the form 

QiA^+QaA^+.-. + ^A^ (5) 

The coefficients Qi, Q 2 , .-.Q n are then called the 'normal components of 
disturbing force.' 

In the application to infinitely small motions Lagrange's equations 

d dT dT dV ~ r , /ax 

take the form 

dir'qi + a 2r q 2 + ... + c lr q-i -I- c 2r q2 + ... - Qr (7) 

or, in the case of normal co-ordinates, 

a r q r + c r q r =Q r (8) 

It is easily seen from this that the dynamical characteristics of the normal 

co-ordinates are (1°) that an impulse of any normal type produces an initial 

motion of that type only, and (2°) that a steady disturbing force of any type 

maintains a displacement of that type only. 

To obtain the free motions of the system we put Q r = 0. Solving (8), we 

find 

q r — A r cos (cr r t + e r ), (9) 

where o~r = {o r ja r )^, (10) 

and A r , € r are arbitrary constants f. Hence a mode of free motion is possible 

in which any normal co-ordinate q r varies alone, and the motion of any particle 

of the system, since it depends linearly on q ry will be simple-harmonic, of 

period 27r/o- r ; moreover the particles will keep step with one another, passing 

simultaneously through their equilibrium positions. The several modes of 

this character are called the 'normal modes' of vibration of the system; their 

* The algebraic proof of this involves the assumption that one at least of the functions T, V 
is essentially positive. In the present case T of course fulfils this condition. 

t The ratio c\1-ir measures the 'frequency' of the oscillation. It is convenient to have a 
name for the quantity a itself; the term 'speed' has been used in this sense by Kelvin and 
G. H. Darwin in their researches on the Tides. 



252 Tidal Waves [chap, vm 

number is equal to that of the degrees of freedom, and any free motion what- 
ever of the system may be obtained from them by superposition, with a proper 
choice of the 'amplitudes' (A r ) and 'epochs' (e r ). It is seen from (10) that in 
any normal mode the mean values (with respect to time) of the kinetic and 
potential energies are equal. 

In certain cases, viz. when two or more of the free periods (27r/a) of the 
system are equal, the normal co-ordinates are to a certain extent indeterminate, 
i.e. they can be chosen in an infinite number of ways. By compounding the 
corresponding modes, with arbitrary amplitudes and epochs, we obtain a small 
oscillation in which the motion of each particle is the resultant of simple- 
harmonic vibrations in different directions, and is therefore, in general, elliptic- 
harmonic, with the same period. This is exemplified in the spherical pendulum ; 
an important instance in our own subject is that of progressive waves in deep 
water (Chapter IX.). 

If any of the coefficients of stability (c r ) be negative, the value of a r is a pure 
imaginary. The circular function in (9) is then replaced by real exponentials, 
and an arbitrary displacement will in general increase until the assumptions 
on which the approximate equation (8) is based becomes untenable. The 
undisturbed configuration is then reckoned as unstable. The necessary and 
sufficient condition of stability (in the present sense) is that the potential 
energy V should be a minimum in the configuration of equilibrium. 

To find the effect of disturbing forces, it is sufficient to consider the case 
where Q r varies as a simple-harmonic function of the time, say 

Q r = C r cos (at + e), (11) 

where the value of a is now prescribed. Not only is this the most interesting 
case in itself, but we know from Fourier's Theorem that, whatever the law of 
variation of Q r with the time, it can be expressed by a series of terms such as 
(11) A particular integral of (8) is then 

&"* ^7 cos(cr* + e) (12) 

This represents the 'forced oscillation' due to the periodic force Q r . In it the 
motion of every particle is simple-harmonic, of the prescribed period 27r/a, 
and the extreme displacements coincide in time with the maxima and minima 
of the force. 

A constant force equal to the instantaneous value of the actual force (11) 
would maintain a displacement 

Q 

q r = -1 cos (at + e), (13) 

c r 

the same, of course, as if the inertia-coefficient a r were null. Hence (12) may 
be written 

q ^T^}^ (14) 



168] Forced Oscillations 253 

where g t has the value (10). This very useful formula enables us to write 
down the effect of a periodic force when we know that of a steady force of the 
same type. It is to be noticed that q r and Q r have the same or opposite phases 
according as cr $ cr r , that is, according as the period of the disturbing force is 
greater or less than the free period. A simple example of this is furnished by 
a simple pendulum acted on by a periodic horizontal force. Other important 
illustrations will present themselves in the theory of the tides *. 

When g is very great in comparison with a r , the formula (12^ becomes 

?r = -^cos(<j£ + e); (15) 

the displacement is now always in the opposite phase to the force, and depends 

only on the inertia of the system. 

If the period of the impressed force be nearly equal to that of the normal 

mode of order r, the amplitude of the forced oscillation, as given by (14), is 

very great compared with q r . In the case of exact equality, the solution (12) 

fails, and must be replaced by 

C t 
gv = 2^- sin (o-£ + e) (16) 

This gives an oscillation of continually increasing amplitude, and can therefore 
only be accepted as a representation of the initial stages of the disturbance. 

Another very important property of the normal modes may be noticed. If by the 
introduction of frictionless constraints the system be compelled to oscillate in any other 
prescribed manner, the configuration at any instant can be specified by one variable, which 
we will denote by 6. In terms of this we shall have 

q r =B r 6, 
where the quantities B r are certain constants. This makes 

2T=(B 1 *a 1 +B 2 *a 2 +...+B n *a n )0*, (17) 

2F=(^ Cl+ ^ C2 + ...+42 c ^2 (18) 

If a cos (o-^ + e), the constancy of the energy (T+ V) requires 

i _ B 1 *c 1 + Bfc a +...+B n *c n . . 

°" B 1 *a 1 + B 2 *a 2 + ...+B 1 ?a n ^ 

Hence o- 2 is intermediate in value between the greatest and least, of the quantities cr/a r ; 
in other words, the frequency of the constrained oscillation is intermediate between the 
greatest and least frequencies corresponding to the normal modes of the system. In par- 
ticular, when a system is modified by the introduction of a constraint, the frequency of 
the slowest natural oscillation is increased. Moreover, if the constrained type differ but 
slightly from a normal type (r), o- 2 will differ from c r /a r by a small quantity of the second 
order. This gives a method of estimating approximately the frequency in cases where the 
normal types cannot be accurately determined f. Examples will be found in Arts. 191, 259. 

* Cf. T. Young, "A Theory of Tides," Nicholson's Journal, xxxv. (1813) [Miscellaneous 
Works, London, 1854, ii. 262]. 

t Eayleigh, "Some General Theorems relating to Vibrations," Proc. Lond. Math. Soc. iv. 
357 (1874) [Papers, i. 170], and Theory of Sound, c. iv. The method was elaborated by Eitz, 
Journ. fur Math, exxxv. 1 (1908), and Ann. der Physik, xxviii. (1909) [Gesammelte Werke, Paris, 
1911, pp. 192, 265]. 



254 Tidal Waves [chap, fiii 

It may further be shewn that in the case of a partial constraint, which merely reduces 
the degree of freedom from n to n — 1, the periods of the modified system separate those of 
the original one* 

It had been already remarked by Lagrange f that if in the equations of type (7), where 
the co-ordinates are not assumed to be normal, we put Q r =0, and assume 

q r = A r cos (o-t + e), (20) 

the resulting equations are identical with those which determine the stationary values of 
the expression 

9 ^ c n AS+c 22 A 2 *+ ... +gci^M2+ ... V(A,-A) . v 

a a n A l *+a 22 A 2 *+...+2a l2 A l A 2 + ..: T (A, A)' l ; 

say. Since T(A, A) is essentially positive the denominator cannot vanish, and the 
expression has therefore a minimum value. It is moreover possible, starting from this 
property, to construct a proof that the n values of o- 2 are all real % . They are obviously all 
positive if V be essentially positive. 

Rayleigh's theorem is also closely related to the Hamiltonian formula (3) of Art. 135, 
as we may see by assuming 

q r =A r sin at, (22) 

and taking t =0, ^— 2jt/ot. Cf. Art. 205 a. 

The modifications which are introduced into the theory of small oscillations 

by the consideration of viscous forces will be noticed in Chapter XL 

Long Waves in Canals. 

169. Proceeding now to the special problem of this chapter, let us begin 
with the case of waves travelling along a straight canal, with horizontal bed, 
and parallel vertical sides. Let the axis of x be parallel to the length of the 
canal, that of y vertical and upwards, and let us suppose that the motion 
takes place in these two dimensions x, y. Let the ordinate of the free surface, 
corresponding to the abscissa x, at time t, be denoted by y + tj, where y is 
the ordinate in the undisturbed stalfe. 

As already indicated, we shall assume in all the investigations of this 
Chapter that the vertical acceleration of the fluid particles may be neglected, 
or, more precisely, that the pressure at any point (x, y) is sensibly equal to 
the statical pressure due to the depth below the free surface, viz. 

p-po = gp(yo + y-y), (i) 

where p is the (uniform) external pressure. 

S-»S < 2 > 

This is independent of y, so that the horizontal acceleration is the same for 
all particles in a plane perpendicular to x. It follows that all particles which 
once lie in such a plane always do so ; in other words, the horizontal velocity 
u is a function of x and t only. 

**Kouth, Elementary Rigid Dynamics, Art. 67; Rayleigh, Theory of Sound (2nd ed.), Art. 92 a; 
Whittaker, Analytical Dynamics, Art. 81. 

t Mecanique Analytique (Bertrand's ed.), i. 331; Oeuvres, xi. 380. 

% See Poincare, Journ. de Math. (5), ii. 83 (1896); Lamb, Higher Mechanics, 2nd ed., Art. 92. 



168-169] Waves in Uniform Canal 255 

The equation of horizontal motion, viz. 

du du _ Idp 
dt dx pdx' 

is further simplified in the ca3e of infinitely small motions by the omission of 
the term udu/dx, which is of the second order, so that 

S"—'S (3) 

Now let S^fudt; 

i.e. £ is the time-integral of the displacement past the plane x, up to the 
time t. In the case of small motions this will, to the first order of small 
quantities, be equal to the displacement of the particles which originally 
occupied that plane, or again to that of the particles which actually occupy it 
at time t The equation (3) may now be written 

*f— ,*S (4) 

The equation of continuity may be found by calculating the volume of 
fluid which has, up to time t, entered the space bounded by the planes x and 
x + Bx; thus, if h be the depth and b the breadth of the canal, 

— a~ (£hb) 8® — rjbSx, 

*— *i (5) 

The same result comes from the ordinary form of the equation of continuity, viz. 

te + ty = ° (6) 



•(7) 



__ fvdu , du 

Thus '--./.B*— »» 

if the origin be (for the moment) taken in the bottom of the canal. This formula is of 
interest as shewing, as a consequence of our primary assumption, that the vertical velocity 
of any particle is simply* proportional to its height above the bottom. At the free surface 
we have y = h + rj, v = drj/dt, whence (neglecting a product of small quantities) 

di-- h fadl (8) 

From this (5) follows by integration with respect to t. 

Eliminating tj between (4) and (5), we obtain 

dt*~ g dx* 

The elimination of f gives an equation of the same form, viz. 

w*= gh w 

The above investigation can readily be extended to the case of a uniform 



3-*s » 



sf'-"3 <>») 



256 Tidal Waves [chap, vin 

canal of any form of section *. If the sectional area of the undisturbed fluid 
be S, and the breadth at the free surface b, the equation of continuity is 

-^S)8x = v bhx, (11) 

whence rj = — h^, (12) 

as before, provided h — S/b, i.e. h now denotes the mean depth of the canal. 
The dynamical equation (4) is of course unaltered. 

170. The equation (9) is of a well-known type which occurs in several 
physical problems, e.g. the transverse vibrations of strings, and the motion of 
sound-waves in one dimension. 

To integrate it, let us write, for shortness, 

«-#) fltt 

and x — ct = x 1} x-\-ct — x^. 

In terms of x± and x z as independent variables, the equation takes the form 

The complete solution is therefore 

£ = F(x-ct)+f(x + ct), (14) 

where F, f are arbitrary functions. 

The corresponding values of the particle- velocity and of the surface-elevation 
are given by 

\ 

^-F'{x-ct)-f'{x + ct). 

The interpretation of these results is simple. Take first the motion repre- 
sented by the first term in (14), alone. Since F (x — ct) is unaltered when t 
and x are increased by t and ct, respectively, it is plain that the disturbance 
which existed at the point x at time t has been transferred at time t + t to 
the point x + ct. Hence the disturbance advances unchanged with a constant 
velocity c in space. In other words we have a 'progressive wave' travelling 
with velocity c in the direction of ^-positive. In the same way the second 
term of (14) represents a progressive wave travelling with velocity c in the 
direction of ^-negative. And it appears, since (14) is the complete solution 
of (9), that any motion whatever of the fluid, which is subject to the conditions 
laid down in the preceding Art., may be regarded as made up of waves of these 
two kinds. 

* Kelland, Trans. B. S. Edin. xiv. (1839). 



c 



.(15) 



169-171] 



Initial Conditions 



257 



The velocity (c) of propagation is, by (13), that 'due to' half the depth of 
the undisturbed fluid*. 

The following table giving, in round numbers and assuming <7=32f/s, the velocity of 
wave-propagation for various depths, will be of interest later in connection with the theory 
of the tides. 

The last column gives the time a wave would take to travel over a distance equal to 
the earth's circumference (%ira). In order that a 'long' wave should traverse this distance 
in 24 hours, the depth would have to be about 14 miles. It must be borne in mind that 
these numerical results are only applicable to waves satisfying the conditions above 
postulated. The meaning of these conditions will be examined more particularly in 
Art. 172. 



h 


c 


c 


2ira/c 


(feet) 


(feet per sec.) 


(sea-miles per hour) 


(hours) 


3121 


100 


60 


360 


1250 


200 


120 


180 


5000 


400 


240 


90 


11250f 


600 


360 


60 


20000 


800 


480 


45 



171. To trace the effect of an arbitrary initial disturbance, let us suppose 
that when t = we have 

\=$(oo), j-f(*j (16) 

The functions F', f which occur in (15) are then given by 

*"(•) — *{*(•) + + (•)},} Q7 x 

/(*>- *{*(»)-*(•)}.; v ' 

Hence if we draw the curves y — rji, y = 7] 2 , where 

i?i = iM* (*) + *(*)}») (1S) 

% = iM*(aO -*(*)},} 

the form of the wave-profile at any subsequent instant t is found by displacing 
these curves parallel to x, through spaces ± ct, respectively, and adding (alge- 
braically) the ordinates. If, for example, the original disturbance is confined 
to a length I of the axis of oo, then after a time l/2c it will have broken up into 
two progressive waves of length I, travelling in opposite directions. 

In the particular case where in the initial state f = 0, and therefore cj> (a?) = 0, 
we have t) X = yz] the elevation in each of the derived waves is then exactly half 
what it was, at corresponding points, in the original disturbance. 

It appears from (16) and (17) that if the initial disturbance be such that 
? = ± v/h • c > the motion will consist of a wave system travelling in one 
direction only, since one or other of the functions F' and f is then zero. 

* Lagrange, Nouv. mem. de V Acad, de Berlin, 1781 [Oeuvres, i. 747]. 

t This is probably comparable in order of magnitude with the mean depth of the ocean. 



258 Tidal Waves [chap, vin 

It is easy to trace the motion of a surface-particle as a progressive wave of 
either kind passes it. Suppose, for example, that 

%=*F(w-ct), (19) 

and therefore %~ c h (^ 

The particle is at rest until it is reached by the wave ; it then moves forward 
with a velocity proportional at each instant to the elevation above the mean 
level, the velocity being in fact less than the wave-velocity c, in the ratio of 
the surface-elevation to the depth of the water. The total displacement at 
any time is given by 



Z=l\ v cdt. 



p Di := -dy- 9p ' 



This integral measures the volume, per unit breadth of the canal, of the 
portion of the wave which has up to the instant in question passed the 
particle. Finally, when the wave has passed away, the particle is left at rest 
in advance of its original position at a distance equal to the total volume of 
the elevated water divided by the- sectional area of the canal. 

172. We can now examine under what circumstances the solution expressed 
by (14) will be consistent with the assumptions made provisionally in Art. 169. 

The exact equation of vertical motion, viz. 

Dv _ dp 

gives, on integration with respect to y, 

p-po=gp(yo + v-y)-p) jy t dy (21) 

This may be replaced by the approximate equation (1), provided fih be small 
compared with gi), where /3 denotes the maximum vertical acceleration. Now 
in a progressive wave, if X denote the distance between two consecutive 
nodes {i.e. points at which the wave-profile meets the undisturbed level), the 
time which the corresponding portion of the wave takes to pass a particle is 
X/c, and therefore, provided the gradient dr)/d% is everywhere small, the vertical 
velocity will be of the order rjc/X*, and the vertical acceleration of the order 
?7C 2 /X 2 , where j] is the maximum elevation (or depression). Hence /3h will be 
small compared with grj, provided h 2 /\ 2 is a small quantity. 

Waves whose slope is gradual, and whose length X is large compared with 
the depth h of the fluid, are called 'long' waves. 

Again, the restriction to infinitely small motions, made in equation (3), 
consisted in neglecting udu/dx in comparison with du/dt. In a progressive 

* Hence, comparing with (20), we see that the ratio of the maximum vertical to the maximum 
horizontal velocity is of the order hj\. 



171-173] Airy's Method 259 

wave we have du/dt = + cdu/dx; so that u must be small compared with c, and 
therefore, by (20), r) must be small compared with h. It is to be observed 
that this condition is altogether distinct from the former one, which may be 
legitimate in cases where the motion cannot be regarded as infinitely small. 
See Art. 187. 

The preceding conditions will of course be satisfied in the general case 
represented by equation (14), provided they are satisfied for each of the two 
progressive waves into which the disturbance can be analysed. 

173. There is another, although on the whole a less convenient, method 
of investigating the motion of 'long' waves, in which the Lagrangian plan is 
adopted of making the co-ordinates refer to the individual particles of the 
fluid. For simplicity, we will consider only the case of a canal of rectangular 
section*. The fundamental assumption that the vertical acceleration may be 
neglected implies as before that the horizontal motion of all particles in a plane 
perpendicular to the length of the canal will be the same. We therefore denote 
by x + f the abscissa at time t of the plane of particles whose undisturbed 
abscissa is x. If rj denote the elevation of the free surface, in this plane, the 
equation of motion of unit breadth of a stratum whose thickness (in the un- 
disturbed state) is Bx will be 

where the factor (dp/dx) . Bx represents the pressure-difference for any two 
opposite particles x and x + Bx on the two faces of the stratum, while the 
factor h + rj represents the area of the stratum. Since we assume that the 
pressure about any particle depends only on its depth below the free surface 
we may write 

dp _ drj 

so that our dynamical equation is 

3--»(>+-9& » 

The equation of continuity is obtained by equating the volumes of a stratum, 
consisting of the same particles, in the disturbed and undisturbed conditions 
respectively, viz. 



Bx + ^- Bx ) (h + v) = hBx, 






(■♦a: < 2 > 



* Airy, Encyc. Metrop. "Tides and Waves," Art. 192 (1845); see also Stokes, "On Waves," 
Camb. and Dub. Math. Journ. iv. 219 (1849) [Papers, ii. 222]. The case of a canal with sloping 
sides has been treated by McCowan, "On the Theory of Long Waves...," Phil. Mag. (5), xxxv. 
250(1892). 



260 Tidal Waves [chap, viii 

Between equations (1) and (2) we may eliminate either rj or f ; the result in 
terms of f is the simpler, being 

ay- 

8 2 f , da? 



dt 2 



gh-, ^r 3 (3) 



('♦B 



This is the general equation of 'long' waves in a uniform canal with vertical 
sides *. 

So far the only assumption in the present investigation is that the vertical 
acceleration of the particles may be neglected in calculating the pressure. If 
we now assume, in addition, that rj/h is a small quantity, the equations (2) 
and (3) reduce to 

?— *S« w 

and S=^S < 5) 

The elevation rj now satisfies an equation of the same form, viz. 

w =gh w (6) 

These are in conformity with our previous results; for the smallness of 
5f /3a? means that the relative displacement of any two particles is never more 
than a minute fraction of the distance between them, so that (to a first ap- 
proximation) it is now immaterial whether the variable x be supposed to refer 
to a plane fixed in space, or to one moving with the fluid. 

174. The potential energy of a wave, or system of waves, due to the 
elevation or depression of the fluid above or below the mean level is, per unit 
breadth, gpjfydxdy, where the integration with respect to y is to be taken 
between the limits and rj, and that with respect to x over the whole length 
of the waves. Effecting the former integration, we get 

igpfv 2 dx (1) 

The kinetic energy is \ph^dx (2) 

In a system of waves travelling in one direction only we have 

so that the expressions (1) and (2) are equal; or the total energy is half 
potential, and half kinetic. 

This result may be obtained in a more general manner, as follows js Any 
progressive wave may be conceived as having been originated by the splitting 

* Airy, I.e. 

t Bayleigh, "On Waves," Phil. Mag. (5), i. 257 (1876) [Papers, i. 251]. 



173-175] Energy of Long Waves 261 

up, into two waves travelling in opposite directions, of an initial disturbance 
in which the particle-velocity was everywhere zero, and the energy therefore 
wholly potential. It appears from Art. 171 that the two derived waves are 
symmetrical in every respect, so that each must contain half the original store 
of energy. Since, however, the elevation at corresponding points is for each 
derived wave exactly half that of the original disturbance, the potential energy 
of each will by (1) be one-fourth of the original store. The remaining (kinetic) 
part of the energy of each derived wave must therefore also be one-fourth of 
the original quantity. 

175. If in any case of waves travelling in one direction only, without 
change of form, we impress on the whole mass a velocity equal and opposite 
to that of propagation, the motion becomes steady, whilst the forces acting on 
any particle remain the same as before. With the help of this artifice, the laws 
of wave-propagation can be investigated with great ease *. Thus, in the present 
case we shall have, by Art. 22 (5), at the free surface, 

^ = const. -#(/* + 77) -l^ 2 , (1) 

where q is the velocity. If the slope of the wave-profile be everywhere gradual, 
and the depth h small compared with the length of a wave, the horizontal 
velocity may be taken to be uniform throughout the depth, and approximately 
equal to q. Hence the equation of continuity is 

q(h + v ) = ch, (2) 

c being the velocity, in the steady motion, at places where the depth of the 
stream is uniform and equal to h. Substituting for q in (1), we have 

V 



£ = const. -^(l + Q-ic^l + D (3) 



Hence if rj/h be small, the condition for a free surface, viz. p = const., is 
satisfied approximately, provided 

c 2 = gh, (4) 

which agrees with our former result. 

The present method also accounts very simply for the relation between 
particle-velocity and surface-elevation already found in Art. 171. From (2) we 
have, approximately, 

r'^-S < 5 > 

Hence in the wave-motion the particle-velocity relative to the undisturbed 
water is crj/h in the direction of propagation. 

When the elevation rj, though small compared with the wave-length, is not 

* Eayleigh, I.e. 



262 Tidal Waves [chap, viii 

regarded as infinitely small, a closer approximation to the wave- velocity is 
secured if in (4) we replace h by ij + h. This gives a wave- velocity 

approximately, where c = \/(gh)> relative to the fluid in the immediate neigh- 
bourhood. Since this fluid has itself a velocity c r]/h, the velocity of propagation 
in space is approximately 

<*4l)> w 

a result due substantially to Airy*. It follows that a wave of the type now 
under consideration cannot be propagated entirely without change of profile, 
since the speed varies with the height. Another proof of (6) will be given 
presently when we come to consider specially the theory of waves of finite 
amplitude (Art. 187). 

176. It appears from the linearity of the approximate equations that, in 
the case of sufficiently low waves, any number of independent solutions may 
be superposed. For example, having given a wave of any form travelling in 
one direction, if we superpose its image in the plane x = 0, travelling in the 
opposite direction, it is obvious that in the resulting motion the horizontal 
velocity will vanish at the origin, and the circumstances are therefore the 
same as if there were a fixed barrier at this point. We can thus understand 
the reflection of a wave at a barrier; the elevations and depressions are 
reflected unchanged, whilst the horizontal velocity is reversed. The same 
results follow from the formula 

%=F(ct-x)-F(ct + x), (1) 

which is evidently the most general value of f subject to the condition that 
£=0 for # = 0. 

We can further investigate without much difficulty the partial reflection of a wave at a 
point where there is an abrupt change in the section of the canal. Taking the origin at 
the point in question, we may write, for the negative side, 

*-*KM'+3. -^KH'K) (2) 

and for the positive side 

*-♦(«-£), «.-« ♦(<-£), (3) 

where the function F represents the original wave, and /, cf> the reflected and transmitted 
portions respectively. The constancy of mass requires that at the point x=Q we should 
have b-Ji x U\ = b 2 h 2 u 2 , where b u b 2 are the breadths at the surface, and A 1? h 2 are the mean 
depths. We must also have at the same point ^ — r]^ on account of the continuity of 
pressure t. These conditions give 

b ^{F(t)-f(t)}J-^cf>(t\ F(t)+f(t)=4>(t\ 
c x c 2 

* "Tides and Waves," Art. 208. 

f It will be understood that the problem admits only of an approximate treatment, on account 
of the rapid change in the character of the motion near the point of discontinuity. The nature 



175-177] Forced Waves 263 

We thence find that the ratios of the elevations in corresponding parts of the reflected and 
incident waves, and of the transmitted and incident waves, are 

f ^ b 1 c 1 -b 2 c 2 <f>__ 26^! 

F &1C1+&2V P b 1 ci + b 2 c 2 1 

respectively. The reader may easily verify that the energy contained in the reflected and 
transmitted waves is equal to that of the original incident wave. 

177. Our investigations, so far, relate to cases of free waves. When, in 
addition to gravity, small disturbing forces X, Y act on the fluid, the equation 
of motion is obtained as follows. 

We assume that within distances comparable with the depth h these forces 
vary only by a small fraction of their total value. On this understanding we 
have, in place of Art. 169 (1), 

P-=-£° = (g-Y)(y n + v -y), (1) 

and theretore i| = (gr - Y) g - (y, + r, - y) g". 

We assume that Y is small compared with g, and (for the reason just stated) 
that hd Yfbx is small compared with X. Hence, with sufficient approximation, 
the equation of horizontal motion , viz. 

9~it*' » 

reduces to the form 

dt*— g dx + X > ' (3) 

where, moreover, X may be regarded as a function of x ana t only. The equation 
of continuity is the same as in Art. 169, viz. 



Hence, on elimination of rj, 



*--*8 <*> 



J-ghJ + x (5) 



The horizontal component of the disturbing force is alone important. 

If the disturbing influence consists of a variable surface -pressure (p ), the 
equation (3) is replaced by 

8t 2 9 dx p dx' K ' 

of the approximation implied in the above assumptions will become more evident if we suppose 
the suffixes to refer to two sections S 1 and S 2 , one on each side of the origin 0, at distances from 
which, though very small compared with the wave-length, are yet moderate multiples of the 
transverse dimensions of the canal. The motion of the fluid will be sensibly uniform over each 
of these sections, and parallel to the length. The condition in the text then expresses that there 
is no sensible change of level between S 1 and S 2 . 



264 Tidal Waves [chap, vm 

whilst (4) is unaltered. In the case of a travelling pressure, say 

p j=f(Ut-x\ (7) 

we find 

h-p(U*-gh) w 

The surface depression is in the same phase with the pressure, or the opposite, 
according as U > *J(gh). 

On the other hand, when it is the bottom which is disturbed, we have 
X = in (2), whilst the equation of continuity becomes 

"-"»=->>!' w 

where r) is the elevation of the bottom above the mean level. Thus in the case 
of a seismic wave 

Vo =f(Ut-x), (10) 

we find 

178. The oscillations of water in a canal of uniform section, closed at both 
ends, may, as in the corresponding problem of Acoustics, be obtained by super- 
position of progressive waves travelling in opposite directions. It is more 
instructive, however, with a view to subsequent more difficult investigations, 
to treat the problem as an example of the general theory sketched in Art. 168. 

We have to determine f so as to satisfy 

W~ (? d^ + X ' (1) 

together with the terminal conditions that f = for x = and x = I, say. 
To find the free oscillations we put X — 0, and assume that 

f oc cos {at + e), 
where a is to be found. On substitution we obtain 

g+S*-* w 

whence, omitting the time-factor, 

„ . . ax , D ax 

t = A sin V B cos — . 

* c c 

The terminal conditions give B=0, and 

al/c = rir, (3) 

where r is integral. Hence the normal mode of order r is given by 

f =^4 r sin -^-cos f— j- +e r J, (4) 

where the amplitude A r and epoch e r are arbitrary. 



177-179] Waves in a Finite Canal 265 

In the slowest oscillation (r = 1), the water sways to and fro, heaping 
itself up alternately at the two ends, and there is a node at the middle 
(x = \l). The period (21 /c) is equal to the time a progressive wave would 
take to traverse twice the length of the canal. 

The periods of the higher modes are respectively J, J, J, ... of this, but 
it must be remembered, in this and in other similar problems, that our theory 
ceases to be applicable when the length Ijr of a semi-undulation becomes 
comparable with the depth h. 

On comparison with the general theory of Art. 168, it appears that the 
normal co-ordinates of the present system are quantities q lt q 2} ... q n such 
that when the system is displaced according to any one of them, say q r , we 
have 

f = q r sin -j- ; 

and we infer that the most general displacement of which the system is 
capable (subject to the conditions presupposed) is given by 

y v . V7TX , 

f = %.sin-£- , (5) 

where q lt q 2 , ... q n are arbitrary. This is in accordance with Fourier's 
Theorem. 

When expressed in terms of the normal velocities and the normal co-ordi- 
nates, the expressions for T and V must reduce to sums of squares. This is 
easily verified, in the present case, from the formula (5). Thus if S denote the 
sectional area of the canal, we find 

2T=o8 f ?dx = 2,a r q 2 , 2V = go f [ r ) 2 dx = $c r q 2 , (6) 

Jo hJo 

where a r = ^pSl, c r = %r 2 7r 2 gphS/l (7) 

It is to be noted that, on the present reckoning, the coefficients of stability 
(cv) increase with the depth. 

Conversely, if we assume from Fourier's Theorem that (5) is a sufficiently 
general expression for the value of f at any instant, the calculation just 
indicated shews that the coefficients q r are the normal co-ordinates; and the 
frequencies can then be found from the general formula (10) of Art. 168; viz. 
we have 

<rr = (Cr/ar)* = rTr(gh)t/l, (8) 

in agreement with (3). 

179. As an example of forced waves we take the case of a uniform hori- 
zontal force 

X=/cos(o-£ + e) (9) 

This will illustrate, to a certain extent, the generation of tides in a land- 
locked sea of small dimensions. 



266 Tidal Waves [chap, viii 

Assuming that f varies as cos {at 4- e), and omitting the time-factor, the 
equation (1) becomes 



the solution of which is 



9^ c 2 * c 2 ' 

f = -4 + 2)sin— + #008 — (10) 

a 2 c c 



The terminal conditions give 

E=l, 2>si^ = (l-co^)/ (11) 

Hence, unless sin al/c = 0, we have D =fja 2 . tan al/2c, so that 



.(12) 



f. 2/ . ax . a- (I — x) , . 

f - ? cos 4^/o) Sm 23 Sm "V- • ° 0S ^ + e) ' 

j hf . a{oc — hl) , , 

and 7) = £= — — - sm — — . cos (at + e). 

crc cos (^ aijC) c 

If the period of the disturbing force be large compared with that of the 
slowest free mode, al/2c will be small, and the formula for the elevation 
becomes 

7i = *-(x-kl)coa(<rt+e) t (13) 

approximately, exactly as if the water were devoid of inertia. The horizontal 
displacement of the water is always in the same phase with the force, so long 
as the period is greater than that of the slowest free mode, or crl/c < it. If the 
period be diminished until it is less than the above value, the phase is 
reversed. 

When the period is exactly equal to that of a free mode of odd order 
(r = 1, 3, 5, ...), the above expressions for f and tj become infinite, and the 
solution fails. As pointed out in Art. 168, the interpretation of this is that, 
in the absence of dissipative forces, the amplitude of the motion becomes so 
great that our fundamental approximations are no longer justified. 

If, on the other hand, the period coincide with that of a free mode of even 
order (r= 2, 4, 6, ...), we have sin al/c = 0, cos al/c — 1, and the terminal 
conditions are satisfied independently of the value of D. The forced motion 
may then be represented by* 

f = _^ s in 2 ^cos(<7*+e) (14) 

This example illustrates the fact that the effect of a disturbing force may 
sometimes be conveniently calculated without resolving the force into its 
'normal components.' 

* In the language of the general theory, the impressed force has here no component of the 
particular type with which it synchronizes, so that a vibration of this type is not excited at all. 
In the same way a periodic pressure applied at any point of a stretched string will not excite any 
fundamental mode which has a node there» even though it synchronize with it. 



179-180] Canal Theory of the Tides 267 

Another very simple case of forced oscillations, of some interest in connection 
with tidal theory, is that of a canal closed at one end and communicating at 
the other with an open sea in which a periodic oscillation 

7j = a cos (at + e) (15) 

is maintained. If the origin be taken at the closed end, the solution is 
obviously 

COS (ax/c) / . , \ /-./jv 

y = a / 7/ v . cos (o-S + e), (16) 

cos (trl/c) v ' x 

I denoting the length. If al/c be small the tide has sensibly the same amplitude 
at all points of the canal. For particular values of I (determined by cos crl/c = 0) 
the solution fails through the amplitude becoming infinite. 

Canal Theory of the Tides. 

180. The theory of forced oscillations in canals, or on open sheets of water, 
owes most of its interest to its bearing on the phenomena of the tides. The 
'canal theory,' in particular, has been treated very fully by Airy*. We will 
consider a few of the more interesting problems. 

The calculation of the disturbing effect of a distant body on the waters of 
the ocean is placed for convenience in an Appendix at the end of this Chapter. 
It appears that the disturbing effect of the moon, for example, at a point P 
of the earth's surface, may be represented by a potential XI whose approximate 
value is 

= i 3 ^?(i-co#^X (1) 

where M denotes the mass of the moon, D its distance from the earth's centre, 
a the earth's radius, y the 'constant of gravitation,' and SV the moon's zenith 
distance at the place P. This gives a horizontal acceleration d£l/ad^, or 

/sin 2*. (2) 

towards the point of the earth's surface which is vertically beneath the moon, 
where 

/ = f^ (3) 

If E be the earth's mass, we may write g = yE/a 2 } whence 

/-? E (sl 

g -2'E'\D / 

Putting M/E=^ tf a/D = ^, this gives f/g = 8'57 x 10" 8 . When the sun is 

the disturbing body, the corresponding result is/Jg — S'7S x 10 -8 . 

It is convenient, for some purposes, to introduce a linear magnitude H, 

defined by 

H = af/g (4) 

* Encycl. Metrop. "Tides and Waves," Section vi. (1845). Several of the leading features of 
the theory had been made out, by very simple methods, by Young, in 1813 and 1823 [Works, ii. 
262, 291]. 



268 Tidal Waves [chap, viii 

If we put a = 21 x 10 6 feet, this gives, for the lunar tide, H= 1*80 ft., and for 
the solar tide H= > 79 ft. It is shewn in the Appendix that H measures the 
maximum range of the tide, from high water to low water, on the ' equilibrium 
theory/ 

181. Take now the case of a uniform canal coincident with the earth's 
equator, and let us suppose for simplicity that the moon describes a circular 
orbit in the same plane. Let f be the displacement, relative to the earth's 
surface, of a particle of water whose mean position is in longitude (f>, measured 
eastwards from some fixed meridian. If co be the angular velocity of the 
earth's rotation, the actual displacement of the particle at time t will be 
f + acot, so that the tangential acceleration will be d 2 £/dt. If we suppose the 
'centrifugal force' to be as usual allowed for in the value of g, the processes 
of Arts. 169, 177 will apply without further alteration. 

If n denote the angular velocity of the moon westward, relative to the fixed 
meridian*, we may write in Art. 180 (2) 

^ = nt + <f> + e, 

so that the equation of motion is 

P= c2 j^ sin2 ^ + * +e > <*> 

The free oscillations are determined by the consideration that f is necessarily 
a periodic function of (£, its value recurring whenever <£ increases by 27r. It 
may therefore be expressed, by Fourier's Theorem, in the form 

f = 2 (P r cosr(f> + Q r smrcf>) (2) 

o 

Substituting in (1), with the last term omitted, it is found that P r and Q r 
must satisfy the equation 

J2D r 2 r 2 

T*?- (3) 

The motion, in any normal mode, is therefore simple-harmonic, of period 
2irajrc. 

For the forced waves, or tides, we find 



f = -1; nw si°2 (nt + $ + e), (4) 

c 2 H 
whence rj — J-j $—, - % cos 2 (nt + <£ + e) (5) 

c — 7i a 

The tide is therefore semi-diurnal (the lunar day being of course understood), 
and is 'direct' or 'inverted,' i.e. there is high or low water beneath the moon, 
according as c ^ na, in other words according as the velocity, relative to the 

* That is, n = w - n lf if n x be the angular velocity of the moon in her orbit. 



I8O-182] Equatorial Canal 269 

earth's surface, of a point which moves so as to be always vertically beneath 

the moon, is less or greater than that of a free wave. In the actual case of 

the earth we have 

c 2 g h h 

~2 — 2 ~ 2 * ~" — Oil—, 

rfar rra a a 

so that unless the depth of the canal were to greatly exceed such depths as 
actually occur in the ocean, the tides would be inverted. 

This result, which is sometimes felt as a paradox, comes under a general 
principle referred to in Art. 168. It is a consequence of the comparative 
slowness of the free oscillations in an equatorial canal of moderate depth. 
It appears from the rough numerical table on p. 257 that with a depth 
of 11250 feet a free wave would take about 30 hours to describe the earth's 
semi-circumference, whereas the period of the tidal disturbing force is only a 
little over 12 hours. 

The formula (5) is, in fact, a particular case of Art. 168 (14), for it may be 
written 

" = r^v^ (6) 

where rj is the elevation given by the 'equilibrium theory,' viz. 

7}= %Hcos2 (nt + <f> + e), (7) 

and o- = 2n, <r = 2c/a. 

For such moderate depths as 10000 feet and under, n 2 a 2 is large compared 
with gh\ the amplitude of the horizontal motion, as given by (4), is then 
//4n 2 or g/4>n 2 a.H, nearly, being approximately independent of the depth. 
In the case of the lunar tide this amplitude is about 140 feet. The maximum 
elevation is obtained by multiplying by 2h/a; this gives, for a depth of 
10000 feet, a height of only *133 of a foot. 

For greater depths the tides would be higher, but still inverted, until we 
reach the critical depth n 2 a 2 /g, which is about 13 miles. For depths beyond 
this limit, the tides become direct, and approximate more and more to the 
value given by the equilibrium theory *. 

182. The case of a circular canal parallel to the equator can be worked 
out in a similar manner. If the moon's orbit be still supposed to lie in the 
plane of the equator, we find by spherical trigonometry 

cos Sr as sin 6 cos (nt + ft-f e), (1) 

where 6 is the co-latitude, and <j> the longitude. The disturbing force in 
longitude is therefore 

^-5^ = -/sm0sin2(n£ + <£+e) (2) 

c 2 E sin 2 6 
This leads to V = \ # _ n 2 a z sin 2 q cqs 2 (m* + ft + e) (3) 

* Cf. Young, I.e. ante p. 253. 



270 Tidal Waves [chap, viii 

Hence if na > c the tide will be direct or inverted according as sin 6 ^ c/na. 
If the depth be so great that c> na y the tides will be direct for all values of 6. 
If the moon be not in the plane of the equator, but have a co-declination 
A, the formula (1) is replaced by- 
cos S- = cos 6 cos A + sin 6 sin A cos a, (4) 

where a is the hour-angle of the moon from the meridian of P. For simplicity, 
we will neglect the moon's motion in declination in comparison with the earth's 
angular velocity of rotation ; thus we put 

a = nt + <p + e, 
and treat A as constant. The resulting expression for the disturbing force 
along the parallel is found to be 

= ^/i/ — - /cos, # sin 2 A sin (nt + 6 + e) 

a sm dd<f) J v t * 

-/sin0sin 2 Asin2(?tf + + e) (5) 

We thence obtain 

c 2 H 

71 = * c 2 -tt 2 a 2 sin 2 6> sin 26>sin 2A C0S (n * + * + e) 

^ ^JL^ sin2ism2Acos2( ^^ +€) (6) 

The first term gives a 'diurnal' tide of period 2ir/n; this vanishes ana 
changes sign when the moon crosses the equator, i.e. twice a month. The 
second term represents a semi-diurnal tide of period tt/u, whose amplitude is 
now less than before in the ratio of sin 2 A to 1. 

183. In the case of a canal coincident with a meridian we should have 
to take account of the fact that the undisturbed figure of the free surface is 
one of relative equilibrium under gravity and centrifugal force, and is therefore 
not exactly circular. We shall have occasion later on to treat the question of 
displacements relative to a rotating globe somewhat carefully; for the present 
we will assume by anticipation that in a narrow canal the disturbances are 
sensibly the same as if the earth were at rest, and the disturbing body were 
to revolve round it with the proper relative motion. 

If the moon be supposed to move in the plane of the equator, the hour- 
angle from the meridian of the canal may be denoted by nt + e, and if 6 be 
the co-latitude of any point P on the canal, we find 

cos ^ = sin 0.cos(w£ + e) (1) 

The equation of motion is therefore 



dt* ~ G 



c 2 ^ 2 -i/sin20.{l+cos2(n* + e)}. ...(2) 
Solving, we find 



c 2 ff 
V = -i.H r cos20-i -2 2-2 cos2 0- cos2 <y + e ) ( 3 ) 



182-184] Tides in a Finite Canal 271 

The first term represents a permanent change of mean level to the extent 

97 = -i#cos2<9 (4) 

The fluctuations above and below the disturbed mean level are given by the 
second term in (3). This represents a semi-diurnal tide ; and we notice that if, 
as in the actual case of the earth, c be less than na, there will be high water 
in latitudes above 45°, and low water in latitudes below 45°, when the moon 
is in the meridian of the canal, and vice versa when the moon is 90° from that 
meridian. These circumstances would be all reversed if c were greater than na. 

When the moon is not on the equator, but has a given declination, the 
mean level, as indicated by the term corresponding to (4), has a coefficient 
depending on the declination, and the consequent variations in it indicate a 
fortnightly (or, in the case of the sun, a semi-annual) tide. There is also 
introduced a diurnal tide whose sign depends on the declination. The reader 
will have no difficulty in examining these points, by means of the general 
value of ft given in the Appendix. 

184. In the case of a uniform canal encircling the globe (Arts. 181, 182) 
there is necessarily everywhere exact agreement (or exact opposition) of phase 
between the tidal elevation and the forces which generate it. This no longer 
holds, however, in the case of a canal or ocean of limited extent. 

Let us take for instance the case of an equatorial canal of finite length*. 
Neglecting the moon's declination we have, if the origin of time be suitably 
chosen, 

?H^-/ 8in2 <"' + *> w 

with the condition that £ = at the ends, where </> = + a, say. 

If we neglect the inertia of the water the term dPg/dt 2 is to be omitted 
and we find 

f = \J\ \ sin 2nt cos 2a + * CO s 2nt sin 2a - sin 2 (nt + <j>)l . . . .(2) 

Hence v =- ^||==J#jcos 2 (nt + <j>) - 5||? cos 2nt\ , (3) 

where H = af/g, as in Art. 180. This is the elevation on the (corrected) 
'equilibrium' theory referred to in the Appendix to this Chapter. At the 
centre (<£ — 0) of the canal we have 

V = iHcos2nt(l-^) (4) 

If a be small the range is here very small, but there is not a node in the absolute 

* H. Lamb and Miss Swain, Phil. Mag. (6), xxix. 737 (1915). A similar enect of variable 
depth is discussed by Goldsbrough, Proc. Lond. Math. Soc. (2) xv. 64 (1915). 



272 Tidal Waves [chap, viii 

sense of the term. The times of high water coincide with the transits of 
moon and 'anti-moon*/ At the ends <f> = + a we have 

i tt (/-, sin4a\ rt/ , v _l — cos4a . _, , . ) 
7; = J# \ 1 1 1 — J cos 2 (ratf ± a) + - sin 2 (n£ + a) [ 

= %HR cos 2 (nt± a + e ), ;.(5) 

•e r> o ^ sin 4a ^ . 1- cos 4a ,1 

11 it cos2eo = l 7 — ' iioSin2e = 7 (6) 

Here e denotes the hour-angle of the moon W. of the meridian when there is 
high water at the eastern end of the canal, or E. of the meridian when there 
is high water at the western end. When a is small we have 

E =2a, 6 =-i7r + fa, (7) 

approximately. 

When the inertia of the water is taken into account we have 

b 4 (m 2 -l)c 2 L sin4raa l 7 vr / 

— sin 2 (wi — a) sin 2m (</> — a)} , (8) 

where m = na/c. Hence f 



— i-JL 



^7=-f 



m 2 — 1 



cos 2 (wi + 6) — : — - — fsin 2 (nt + a) cos 2m (<f> 4- a) 
^ sm4ma l r 7 

— sin 2 (n£ — a) cos 2m ((/> — a)} . .., (9) 



If we imagine m to tend to the limit we obtain the formula (3) of the 

equilibrium theory. It may be noticed that the expressions do not become 

infinite for m -> 1 as they would in the case of an endless canal. In all cases 

which are at all comparable with oceanic conditi6ns m is, however, considerably 

greater than unity. 

At the centre of the canal we have 

, H ( msin2a\ 

V = - f — 9 — =- cos 2nd 1 r— = I (10) 

' 2 m 2 - 1 V sm 2ma } v ' 

As in the equilibrium theory, the range is very small if a be small, but there 
is not a true node. At the ends we find 

.. H ( /m sin 4a , \ _ . , N 

7 ? = i-2 — T\\-r—A 1 ) cob 2 (n* ± a) 

z m 2 — 1 (\sin4ma / 

a)f 

= iHR 1 cos2(nt±a+ ei ), (11) 

if 

„ _ m sin 4a — sin 4ma „ . a m (cos 4ma — cos 4a) ,,„ 

R 1 cos 2e 2 = — — = — , , R x sin 2e x = -~-^ — , x . . . . . .(12) 

(m 2 - 1) sin 4ma ' (m 2 - 1) sm 4ma v ; 

* This term is explained in the Appendix to this Chapter, 
t Cf. Airy, "Tides and Waves," Art. 301. 



m (cos 4ma — cos 4a) . _ , . 

f — : , — - - sm 2 (nt ± 

sm 4ma 



184-185] 



Tides in Finite Canal 



273 



When a is small we have 

R 1 =2a, 6i=-i7r + fa, (13) 

approximately, as in the case of the equilibrium theory. 

The value of R x becomes infinite when sin 4>ma = 0. This determines the 
critical lengths of the canal for which there is a free period equal to irjn, or 
half a lunar day. The limiting value of ei in such a case is given by 

tan 2ei = — cot 2a, or = tan 2a, 
according as 4ma is an odd or even multiple of it. 





Corrected Equilibrium Theory 


Dynamical Theory 


2a 


2aa 


Kange at 


Eange at 


e o 


Eange at 


Kange at 


«i 


(degrees) 


(miles) 


centre 


ends 


(degrees) 


centre 


ends 


(degrees) 














-45 








-45 


9 


540 


•004 


•157 


-42 


•004 


•165 


-4V9 


18 


1080 


•016 


•311 


-39 


•018 


•396 


-38'5 


27 


1620 


•037 


•460 


-36 


•044 


•941 


-33'9 


31' 5 


1890 


•050 


•531 


-34-5 


•063 


1-945 


-30-9 


36 


2160 


•065 


•601 


-33 


•089 


CO 


f -27 
\ + 63 
+ 68-2 


40 '5 


2430 


•081 


•668 


-31-6 


•125 


1-956 


45 


2700 


•100 


•733 


-30-1 


•174 


•987 


+ 75'7 


54 


3240 


•142 


•853 


-27'2 


•354 


•660 


-83-5 


63 


3780 


•190 


•959 


-24-4 


•918 


1-141 


-65-1 


72 


4320 


•243 


1-051 


-21-6 


CO 


CO 


(-54 

{+36 

+44' 5 


81 


4860 


•301 


1-127 


-18'9 


1-459 


1-112 


90 


5400 


•363 


1-185 


-16'2 


•864 


•513 


+ 55-9 



The table illustrates the case of'ra = 2*5. If 77-/^ = 12 lunar hours this implies a depth 
of 10820 ft., which is of the same order of magnitude as the mean depth of the ocean. 
The corresponding wave-velocity is about 360 sea-miles per hour. The first critical 
length is 2160 miles (a = ^7r). The unit in terms of which the range is expressed is the 
quantity H, whose value for the lunar tide is about 1-80 ft. The hour-angles e and ej 
are adjusted so as to lie always between ±90°, and the positive sign indicates position W. 
of the meridian in the case of the eastern end of the canal, and E. of the meridian for the 
western end. 

Wave-Motion in a Canal of Variable Section. 
185. When the section (S, say) of the canal is not uniform but varies 
gradually from point to point, the equation of continuity is by Art. 169 (11), 



'—!&<«& 



•d) 



where b denotes the breadth at the surface. If h denote the mean depth over 
the width b, we have S = bh, and therefore 



'— js>& 



•(?) 



where h, b are now functions of x. 



274 Tidal Waves [chap, viii 

The dynamical equation has the same form as before, viz. 

dt*~ 9 dx {S) 

Between (2) and (3) we may eliminate either rj or f ; the equation in t) is 

dt 2 bdxV°dx) w 

The laws of propagation of waves in a canal of gradually varying rect- 
angular section were investigated by Green*. His result, freed from the 
restriction to the special form of section, may be obtained as follows. 
If we introduce a variable t denned by 

3f-&*A ■■•<•> 

in place of x, the equation (4) transforms into 

where the accents denote differentiations with respect to r. If b and h were constants, the 

equation would be satisfied by rj = F (r — t), as in Art. 170 ; in the present case we assume 

for trial, 

v =e.F(r-t), (7) 

where 9 is a function of r only. Substituting in (6), we find 



e' F' e" (V l h'\ (F' e'\ rt 



.(8) 



The terms of this which involve F will cancel provided 

a e' , b' U' A 

or e = Cb-$k-±, (9) 

C being a constant. Hence, provided the remaining terms in (8) may be neglected, the 
equation (4) will be satisfied. 

The above approximation is justified, provided we can neglect Q"/Q' and 0'/0 in com- 
parison with F'/F. As regards e'/6, it appears from (9) and (7) that this is equivalent to 
neglecting b~ l .dbjdx and h~ l .dhjdx in comparison with i}' 1 . drj/dx. If, now, X denote a 
wave-length, in the general sense of Art. 172, drj/dx is of the order 77/X, so that the assump- 
tion in question is that Xdb/dx and Xdh/dx are small compared with b and A, respectively. 
In other words, it is assumed that the transverse dimensions of the canal vary only by 
small fractions of themselves within the limits of a wave-length. It is easily seen, in like 
manner, that the neglect of 0"/6' in comparison with F'/F implies a similar limitation to 
the rates of change of dbjdx and dh/dx. 

Since the equation (4) is unaltered when we reverse the sign of £, the complete solution, 
subject to the above restrictions, is 

l-&-*A-t{F(r-0+/(r+0}. (10) 

where F and / are arbitrary functions. 

The first term in this represents a wave travelling in the direction of .^-positive ; the 
velocity of propagation past any point is determined by the consideration that any particular 
phase is recovered when fir and bt have equal values, and is therefore equal to *J(gh), by 

* "On the Motion of Waves in a Variable Canal of small depth and width," Gamb. Trans, vi. 
(1837) [Papers, p. 225]; see also Airy, "Tides and Waves," Art. 260. 



185-186] Canal of Varying Section 275 

(5), as we should expect from the case of a uniform section. In like manner the second 
term in (10) represents a wave travelling in the direction of ^-negative. In each case the 
elevation of any particular part of the wave alters, as it proceeds, according to the law 

The reflexion of a progressive wave at a point where the section of a 
canal suddenly changes has been considered in Art. 176. The formulae there 
given shew, as we should expect, that the smaller the change in the 
dimensions of the section, the smaller will be the amplitude of the reflected 
wave. The case where the change from one section to the other is con- 
tinuous, instead of abrupt, has been investigated by Rayleigh for a special 
law of transition*. It appears that if the space within which the transition 
is completed be a moderate multiple of a wave-length there is practically no 
reflexion; whilst in the opposite extreme the results agree with those of 
Art. 176. 

If we assume, on the basis of these results, that when the change of 
section within a wave-length may be neglected a progressive wave suffers 
no appreciable disintegration by reflexion, the law of amplitude easily follows 
from the principle of energy |. It appears from Art. 174 that the energy of 
the wave varies as the length, the breadth, and the square of the height, and 
it is easily seen that the length of the wave, in different pafts of the canal, 
varies as the corresponding velocity of propagation, and therefore as the square 
root of the mean depth. Hence in the above notation, rfbh^ is constant, or 

which is Green's law above found. 

186. In the case of simple harmonic motion, where t] oc cos {at + e), the 
equation (4) of the preceding Art. becomes 



I&KlH-o a> 



Some particular cases of considerable interest can be solved with ease. 

1°. For example, let us take the case of a canal whose breadth varies as the distance 
from the end # = 0, the depth being uniform ; and let us suppose that at its mouth {x=a) 
the canal communicates with an open sea in which a tidal oscillation 

t) = C cos (<rt + e) (2) 

is maintained. Putting h = const., b <x x, in (1), we find 

S-4l+* 2 -°> v 

provided k 2 = a 2 /gh (4) 

Hence "- c W) coa{,Tt+e) (5) 

* " On Reflection of Vibrations at the Confines of two Media between which the Transition is 
gradual," Proc. Lond. Math. Soc. (1) xi. 51 (1880) [Papers, i. 460]; Theory of Sound, 2nd ed., 
London, 1894, Art. 148 b. 

t Rayleigh, I.e. ante p. 260. 



276 



Tidal Waves 



[CHAP. VIII 



The curve y=«/ i x ) i s figured on p. 286 ; it indicates how the amplitude of the forced 
oscillation increases, whilst the wave-length is practically constant, as we proceed up the 
canal from the mouth. 

2°. Let us suppose that the variation is in the depth only, and that this increases 
uniformly from the end x—Ooi the canal to the mouth, the remaining circumstances 
being as before. If, in (1), we put h — h^x\a, K = (r 2 a/gh 0i we obtain 

s(49+"=o- m 

The solution of this which is finite for #=0 is 

f KOI) IC 3S \ 

7? = ^l- T2 + T 2- y2 -...j, (7) 

or i7 = ^Le/ (2#c*a?*), (8) 

whence finally, restoring the time-factor and determining the constant, 



J (2k* at) 



.(9) 



0- 




The annexed diagram of the curve y = Jo{<Jx\ where, for clearness, the scale adopted 
for y is 200 times that of x, shews how the amplitude continually increases, and the 
wave-length diminishes, as we travel up the canal. 

These examples may serve to illustrate the exaggeration of oceanic tides which takes 
place in shallow seas and in estuaries. 

3°. If the breadth and depth both vary as the distance from the end # = 0, we have, 
writing b = \x\a, h — h x/a, 



dx* 



+ ^ + ^ = 0, 



ox 



.(10) 



where k = a 2 a/gh as before. Hence 

* =A i 1 ~ O + 17170" -) cos (<rt+t) (11) 

The series is equal to J x (2k*#*)/kz#z, and the constant A is determined by com- 
parison with (2). The present assumption gives a fair representation of the case of the 
Bristol Channel, and the tides observed at various stations are found to be in good agree- 
ment with the formula*. 

We add one or two simple problems of free oscillations. 

* G. I. Taylor, Gamb. Proc. xx. 320 (1921). 



186] Canal of Varying Section 277 

4°. Let us take the case of a canal of uniform breadth, of length 2a, whose bed, slopes 
uniformly from either end to the middle. If we take the origin at one end, the motion in 
the first half of the canal will be determined, as above, by 

n =AJ (2 K M), (12) 

where K = o- 2 a/gh , h denoting the depth at the middle. 

It is evident that the normal modes will fall into two classes. In the first of these 77 
will have opposite values at corresponding points of the two halves of the canal, and will 
therefore vanish at the centre (x = a). The values of 0- are then determined by 

</ (2*£a*) = 0, (13) 

viz. < being any root of this, we have 

„J&£.. {Ka) k (14) 

a K 

In the second class, the value of 77 is symmetrical with respect to the centre, so that 
dr)/dx=Q at the middle. This gives 

J '(2 K M)=0 (15) 

It appears that the slowest oscillation is of the asymmetrical class, and corresponds to 
the smallest root of (13), which is 2k* a* = -765577-, whence 

2tt ^ 4a 
— = 1*306 x 5-. 

5°. Again, let us suppose that the depth of the canal varies according to the law 

h = h (l-^, (16) 

where x now denotes the distance from the middle. Substituting in (1), with 6=const., 

we find 

d_ 

dx 



{0-5)&} + »'- (17) 



If we put 0-2 = ^(^+1)^0^ (18) 

this is of the same form as the general equation of zonal harmonics, Art. 84 (1). 

In the present problem n is determined by the condition that 77 must be finite for 
x\a— +1. This requires (Art. 85) that n should be integral; the normal modes are 
therefore of the type 

v =CP n (I) . cos (at + e), (19) 

where P n is a zonal harmonic, the value of o- being determined by (18). 

In the slowest oscillation (n=l), the profile of the free surface is a straight line. For a 
canal of uniform depth k , and of the same length (2a), the corresponding value of <r 
would be 7rc/2a, where c-—(gh )i. Hence in the present case the frequency is less, in the 
ratio 2 ,/2tt, or -9003*. 

The forced oscillations due to a uniform disturbing force 

Xs=/cos(o-* + e) (20) 

* For extensions, and applications to the theory of 'seiches' in lochs, see Chrystal, "Some 
Results in the Mathematical Theory of Seiches," Proc. R. S. Edin. xxv. 328 (1904), and Trans. 
E. S. Edin. xli. 599 (1905). For more recent investigations see Proudman, Proc. Lond. Math. 
Soc. (2) xiv. 240 (1914); Doodson, Trans. R. S. Edin. lii. 629 (1920); Jeffreys, M. N. R. A. S., 
Geophys. Suppt. i. 495 (1928). 



278 Tidal Waves [chap, viii 

can be obtained by the^rule of Art. 168 (14). The equilibrium form of the free surface is 
evidently 

i7 = ^#cos(o-* + e), (21) 



and, since the given force is of the normal type 7i = l, we have 

1 

9^-° 2 l<r<?) 
where o- 2 = 2gh /a 2 . 



9=T7rTb^N* cos (**+«), (22) 



Waves of Finite Amplitude. 

187. When the elevation rj is not small compared with the mean depth 
h, waves, even in an uniform canal of rectangular section, are no longer 
propagated without change of type. The question was first investigated by 
Airy*, by methods of successive approximation. He found that in a pro- 
gressive wave different parts will travel with different velocities, the wave- 
velocity corresponding to an elevation t) being given approximately by Art. 
175(6). 

A more complete view of the matter can be obtained by a method similar 
to that adopted by Riemann in treating the analogous problem in Acoustics. 
(See Art. 282.) 

The sole assumption on which we are now proceeding is that the vertical 
acceleration may be neglected. It follows, as explained in Art. 168, that the 
horizontal velocity may be taken to be uniform over any section of the canal. 
The dynamical equation is 

du du drj /n . 

s* +u to = -^' (1 > 

as before, and the equation of continuity, in the case of a rectangular section, 
is easily seen to be 

!> + '>«> — l! < 2 > 

where h is the depth. This may be written 

_ +u ___ ( A + , )g - (o) 

Multiplying this equation by/' (rj), where /(??) is a function to be deter- 
mined, and adding to (1), we have 

|+«|){/«+«i— (fc+t)/w£-*g 

= _ (h + ,)/(,) A [/(,) + «}, (4) 

provided (* + *) If (I)}*- 0- 

* I.e. ante p. 267. 



186-187] Waves of Finite Amplitude 279 

This is satisfied by 

/(i,)-2*{(l + j[)*-l}, (5) 

where Cq = y/(gh). Hence, writing 

P=f(r,) + u, Q=f( V )-u, (6) 

we have 

f + (" + ">S= ' ••••• w 

and, by similar steps, 

*+<— >g-°. - < 8 > 

where v = (h + v )f\ v ) = c (l + f) ( 9 ) 

It appears, therefore, that P is constant for a geometrical point moving in 
the positive direction of x with the velocity 

c ,(l + ff + u, (10) 

whilst Q is constant for a point moving in the negative direction with the 
velocity 

<*{*+$-« (11) 

Hence any given value of P travels forwards, and any given value of Q travels 
backwards, with the velocities given by (10) and (11) respectively. The 
values of P and Q are determined by those of rj and u, and conversely. 

As an example, let us suppose that the initial disturbance is confined to 
the space for which a < x < b, so that P and Q are initially zero for x < a and 
x > b. The region within which P differs from zero therefore advances, whilst 
that within which Q differs from zero recedes, so that after a time these 
regions separate, and leave between them a space within which P = 0, Q = 0, 
and the fluid is therefore at rest. The original disturbance has now been 
resolved into two progressive waves travelling in opposite directions. 

In the advancing wave we have 

Q = 0, iP=» = 2c„|(l + g i -l} (12) 

so that the elevation ana the particle-velocity are connected by a definite 
relation (cf. Art. 171). The wave-velocity is given by (10) and (12), viz. it is 



Co 



h) 



31+1-2 , (13) 



To the first order ofrj/h, this is in agreement with Airy's result quoted on p. 262. 

Similar conclusions can be drawn in regard to the receding wave*. 

* The above results can also be deduced from the equation (3) of Art. 173, by a method due 
to Earnshaw ; see Art. 283. 



280 Tidal Waves [chap, viii 

Since the wave-velocity increases with the elevation, it appears that in 
a progressive wave-system the slopes will become continually steeper in front, 
and more gradual behind, until at length a state of things is reached in which 
we are no longer justified in neglecting the vertical acceleration. As to what 
happens after this point we have at present no guide from theory; observa- 
tion shews, however, that the crests tend ultimately to curl over and break. 

The case of a ' bore,' where there is a transition from one uniform level to another, 
may be investigated by the artifice of steady motion (Art. 175). If Q denote the volume 
per unit breadth which crosses each section in unit time we have 

u 1 h l = u 2 h 2 =Q, (14) 

where the suffixes refer to the two uniform states, h x and h 2 denoting the depths. Con- 
sidering the mass of fluid which is at a given instant contained between two cross-sections, 
one on each side of the transition, we see that in unit time it gains momentum to the 
amount pQ (u 2 — «i), the second section being supposed to lie to the right of the first. 
Since the mean pressures over the sections are \gph x and \gph 2 , we have 

Q(u 2 -u l ) = y(k 1 *-h 2 *) (15) 

Hence, and from (14), 

£ 2 =%W^i + A 2 ) (16) 

If we impress on everything a velocity — u x we get the case of a wave invading still water 
with a velocity of propagation 

-Vfr} (l7 > 

in the negative direction. The particle-velocity in the advancing wave is % - u 2 in the 
direction of propagation. This is positive or negative according as h 2 ^ h u i.e. according 
as the wave is one of elevation or depression. 

The equation of energy is however violated, unless the difference of level be regarded 
as infinitesimal. If, in the steady motion, we consider a particle moving along the surface 
stream-line, its loss of energy in passing the place of transition is 

\pW-u?)+gp(h,-h 2 ) (18) 

per unit volume. In virtue of (14) and (16) this takes the form 

gp{h 2 -hf 

Ah x h 2 (iy) 

Hence, so far as this investigation goes, a bore of elevation (h 2 > h x ) can be propagated 
unchanged on the assumption that dissipation of energy takes place to a suitable extent 
at the transition. If however h 2 <h x , the expression (19) is negative, and a supply of 
energy would be necessary. It follows that a negative bore of finite height cannot in 
any case travel unchanged*. 

188. In the detailed application of the equations (1) and (3) to tidal 
phenomena, it is usual to follow the method of successive approximation. 
As an example, we will take the case of a canal communicating at one end 
{x = 0) with an open sea, where the elevation is given by 

rj = a cos at (20) 

* Kayleigh, "On the Theory of Long Waves and Bores," Proc. Roy. Soc. A, xc. 324 (1914) 
[Papers, vi. 250]. 



187-188] 



Tides of Second Order 



281 



For a first approximation we have 



du 

dt 



= -9 



h 

dx' 



drj_ ,du 
dt~ dx 



,(21) 



.(22) 



the solution of which, consistent with (20), is 

t) = a cos a- [ t — , u = — i 

For a second approximation we substitute these values of rj and u in (1) and (3), and 
obtain 

du_ drj g 2 aa 2 

dt~~ 9 dx~ ~~ ' 



2c 3 



<-?)• 



'»*'('-!). i-»s-^ ! -«'('-?) : •••« 



Integrating these by the usual methods, we find, as the solution consistent with (20), 



rj = a cos alt— 



9<ra' 
c 3 



x sin 2o- 



(<-?)• 



ga / x\ , a 2 a 2 _ / x\ ..g 2 aa 2 . _ ■/ ■' a?' 

w = ^- cos o-f ? — J- ^^-3-cos2o-( *--)- |^— T -^sin2o- f *-- 



54) 



The annexed figure shews, with, of course, exaggerated amplitude, the profile of the 
waves in a particular case, as determined by the first of these equations. It is to be noted 
that if we fix our attention on a particular point of the canal, the rise and fall of the 
water do not take place symmetrically, the fall occupying a longer time than the rise. 




The occurrence of the factor x outside trigonometrical terms in (24) shews that there is 
a limit beyond which the approximation breaks down. The condition for the success of 
the approximation is evidently that gaax/c 3 should be small. Putting c 2 =gh, X = 27rc/o-, 
this fraction becomes equal to 2n- (a/h) . (x/X). Hance however small the ratio of the 
original elevation (a) to the depth, the fraction ceases to be small when x is a sufficient 
multiple of the wave-length (X). 

It is to be noticed that the limit here indicated is already being overstepped in the 
right-hand portions of the figure; and that the peculiar features which are beginning 
to shew themselves on the rear slope are an indication rather of the imperfections of the 
analysis than of any actual property of the waves. If we were to trace the curve further, 
we should find a secondary maximum and minimum of elevation developing themselves 
on the rear slope. In this way Airy attempted to explain the phenomenon of a double 
high- water which is observed in some rivers; but, for the reason given, the argument 
cannot be sustained*. 

The same difficulty does not necessarily present itself in the case of a canal closed by a 
fixed barrier at a distance from the mouth, or, again, in the case of the forced waves due to 
a periodic horizontal force in a canal closed at both ends (Art. 179). Enough has, however, 
been given to shew the general character of the results to be expected in such cases. For 
further details we must refer to Airy's treatise f. 

When analysed, as in (24), into a series of simple-harmonic functions of the time, the 
expression for the elevation of the water at any particular place (x) consists of two terms? 

* McCowan, I.e. ante p. 259. 
f "Tides and Waves," Arts. 198, .. 
Britann. (9th ed.) xxiii. 362, 363 (1888). 



and 308. See also G. H. Darwin, "Tides," Encyc. 



282 Tidal Waves [chap, vm 

of which the second represents an ' over- tide,' or 'tide of the second order,' being propor- 
tional to a 2 ; its frequency is double that of the primary disturbance (20). If we were to 
continue the approximation we should obtain tides of higher orders, whose frequencies are 
3, 4, ... times that of the primary. 

If, in place of (20), the disturbance at the mouth of the canal were given by 

£ = a cos crt + a' cos (a't + c), 

it is easily seen that in the second approximation we should in like manner obtain tides of 
periods 27rl(cr + a-') and 2n7((r- </) ; these are called 'compound tides.' They are analogous 
to the 'combination- tones' in Acoustics which were first investigated by Helmholtz*. 

Propagation in Two Dimensions. 

189. Let us suppose, in the first instance, that we have a plane sheet 
of water of uniform depth h. If the vertical acceleration be neglected, the 
horizontal motion will as before be the same for all particles in the same 
vertical line. The axes of x, y being horizontal, let u, v be the component 
horizontal velocities at the point (x, y), and let f be the corresponding elevation 
of the free surface above the undisturbed level. The equation of continuity 
may be obtained by calculating the flux of matter into the columnar space 
which stands on the elementary rectangle 8x8y ; thus we have, neglecting 
terms of the second order, 

l x (uUy)^ + l y (vh&x) 8y - -| {(?+ h) 8x8y}, 

whence I=- A (S + |) w 

The dynamical equations are, in the absence of disturbing forces, 

du _ dp dv _ dp 

where we may write 

if zq denote the ordinate of the free surface in the undisturbed state. We 

thus obtain 

du 8f dv d£ /ox 

dt = -!>te> di=- g dy (2) 

If we eliminate u and v, we find 

where c 2 = gk as before. 

In the application to simple-harmonic motion, the equations are shortened 
if we assume a complex time-factor e i{<rt+e) , and reject in the end, the 

t "Ueber Combination stone," Berl. Monatsber. May 22, 1856 [Wiss. Abh. i. 256]; and 
"Theorie der Luftschwingungen in Rohren mit offenen Enden," Crelle, lvii. 14 (1859) [Wiss. 
Abh. i. 318]. 



p dt dx } p dt dy' 



i88-i9o] Waves on an Open Sheet of Water 283 

imaginary parts of our expressions. This is legitimate so long as we have to 
deal solely with linear equations. We have then, from (2), 

u Jlf, v Jlf (4) 

<t dec a- oy 

whilst (3) becomes 

S + p + ^=° (5) 

where A^o^/c 2 (6) 

The condition to be satisfied at a vertical bounding wall is obtained at 
once from (4), viz. it is 

!=°> < 7 > 

if 8n denote an element of the normal to the boundary. 

When the fluid is subject to small disturbing forces whose variation 
within the limits of the depth may be neglected, the equations (2) are 
replaced by 

du = _ 9£_9ft ^ = _ d l_ d & /ft\ 

dt~ 9 dx dx* dt~ 9 dy dy' {) 

where Q is the potential of these forces. 

If we put ?=-0/#, (9) 

so that f denotes the equilibrium-elevation corresponding to the potential XX 
these may be written 

ir-4<^>- l-4 (? - ?) (10) 

In the case of simple-harmonic motion; these take the forms 

-£s«-& -?£«-& • < X1 > 

whence, substituting in the equation of continuity (1) we obtain 

(V 1 2 + ^)ir=v 1 ^ ) (i2) 

if v *-b*k (13) 

and P = a 2 /gh, as before. The condition to be satisfied at a vertical boundary 
is now 

4ff-f>'- < 14) 

190. The equation (3) of Art. 189 is identical in form with that which 
presents itself in the theory of the transverse vibrations of a uniformly 
stretched membrane. A still closer analogy, when regard is had to the 
boundary-conditions, is furnished by the theory of cylindrical waves of 
sound*. Indeed many of the results obtained in this latter theory can be 
at once transferred to our present subject. 

* Eayleigh, Theory of Sound, Art. 338. 



g84 Tidal Waves [chap, viii 

Thus, to find the free oscillations of a sheet of water bounded by vertical 
walls, we require a solution of 

ov+m-o, (i) 

subject to the boundary-condition 

I- » 

Just as in Art. 178 it will be found that such a solution is possible only for 
certain values of k, which accordingly determine the periods (2-Tr/kc) of the 
various normal modes. 

Thus, in the case of a rectangular boundary, if we take the origin at one 
corner, and the axes of x, y along two of the sides, the boundary- conditions 
are that 3f/3# = for x — and x = a, and 3f/3y = for 2/ = and y = b, 
where a, 6 are the lengths of the edges parallel to x, y respectively. The 
general value of f subject to these conditions is given by the double Fourier's 
series 

f=22-4 m>n cos — — cos-yS (3) 

where the summations include all integral values of m, n from to oo . 
Substituting in (1) we find 

»--(3!+-J) < 4 > 

If d > 6, the component oscillation of longest period is got by making m = 1, 
ft = 0, whence ka = tt. The motion is then everywhere parallel to the longer 
side of the rectangle. Cf. Art. 178. 

191. In the case of a circular sheet of water, it is convenient to take the 
origin at the centre, and to transform to polar co-ordinates, writing 

x = r cos 0, y = r sin 6. 

The equation (1) of the preceding Art. becomes 

This might of course have been established independently. 

As regards dependence on 0, the value of f may, by Fourier's Theorem, 
be supposed expanded in a series of cosines and sines of multiples of 0; we 
thus obtain a series of terms of the form 



/(r) C0S l 
J v ; sinj 



•(2) 



It is found on substitution in (1) that each of these terms must satisfy the 
equation independently, and that 



f(T) + lf (r) + (*-J)/(r)-0 (3) 



i90-i9i] Circular Basin 285 

This is of the same form as Art. 101 (14). Since f must be finite for r = 0, 
the various normal modes are given by 

Z=A s J 8 (kr) C °^ s6 .cos(<rt + e), (4) 

where s may have any of the values 0, 1, 2, 3, ..., and A 8 is an arbitrary 
constant. The admissible values of h are determined by the condition that 
d£/dr = at the boundary r = a, say, or 

J s '(ka) = (5) 

The corresponding ' speeds ' (cr) of the oscillations are then given by <r = kc, 
where c = \/(gh). 

In the case 5 = 0, the motion is symmetrical about the origin, so that the 
waves have annular ridges and furrows. The lowest roots of 

J '(&a) = 0, or Ji(A;a) = 0, (6) 

are given by 

ka/7r = 1-2197, 2-2330, 3*2383, ..., (7) 

these numbers tending ultimately to the form &a/7r = ra + £, where m is 
integral *. Hence 

o-a/c=2-832, 7'016, 10*173, (7a) 

In the mth mode of the symmetrical class there are m nodal circles whose 
radii are given by f = or 

J (kr) = (8) 

The roots of this are*f" 

&r/7r=-7655, 1*7571, 2'7546, (9) 

For example, in the first symmetrical mode there is one nodal circle r = *628a. 

The form of the section of the free surface by a plane through the axis of z, 

in any of these modes, will be understood from the drawing of the curve 

y = Jo (#)> which is given on the next page. 

When s > there are s equidistant nodal diameters, in addition to the 

nodal circles 

J s (kr) = (10) 

It is to be noticed that, owing to the equality of the frequencies of the two 

modes represented by (4), the normal modes are now to a certain extent 

indeterminate ; viz. in place of cos sd or sin sd we might substitute cos s (0 — a s ), 

where a s is arbitrary. The nodal diameters are then given by 

a 2m +1 

0-&s= — 27" 71 "' ( n ) 

where ra = 0, 1, 2, ..., 5 — 1. The indeterminateness disappears, and the 
frequencies become unequal, if the boundary deviate, however slightly, from 
the circular form. 

* Stokes, "On the Numericai Calculation of a class of Definite Integrals and Infinite Series." 
Camb. Trans, ix. (1850) [Papers, ii. 355]. 

It is to be noticed that ka/v is equal to tJt, where r is the actual period, and r is the time 
a progressive wave would take to travel with the velocity *J(gh) over a space equal to the 
diameter 2a. f Stokes, I.e. 



286 



Tidal Waves 



[chap, viii 




191J Circular Basin 287 

In the case of the circular boundary, we obtain by superposition of two 
fundamental modes of the same period, in different phases, a solution 

Z=C s J s (kr).cos(crt + s6 + €) (12) 

This represents a system of waves travelling unchanged round the origin 
with an angular velocity ajs in the positive or negative direction of 6. The 
motion of the individual particles is easily seen from Art. 189 (4) to be 
elliptic-harmonic, one principal axis of each elliptic orbit being along the 
radius vector. All this is in accordance with the general theory recapitulated 
in Art. 168. 

The most interesting modes of the unsymmetrical class are those corre- 
sponding to s = 1, e.g. 

^AJiikr) cos 0. cos (crt + e), (13) 

where k is determined by 

J 1 '(ka) = (14) 

The roots of this are * 

kalw= '586, 1-697, 2-717,..., (15) 

whence aa/c =1*841, 5'332, 8536, (15a) 

We have now one nodal diameter (6 = %7r), whose position is, however, in- 
determinate, since the origin of 6 is arbitrary. In the corresponding modes 
for an elliptic boundary, the nodal diameter would be fixed, viz. it would 
coincide with either the major or the minor axis, and the frequencies would 
be unequal. 

The diagrams on the next page shew the contour-lines of the free 
surface in the first two modes of the present species. These lines meet the 
boundary at right angles, in conformity with the general boundary-condition 
(Art. 190 (2)). The simple-harmonic vibrations of the individual particles 
take place in straight lines perpendicular to the contour-lines, by Art. 189 (4). 
The form of the sections of the free surface by planes through the axis of z is 
given by the curve y = Ji {x) on the opposite page. 

The first of the two modes here figured has the longest period of all the 
normal types. In it, the water sways from side to side, much as in the slowest 
mode of a canal closed at both ends (Art. 178). In the second mode there 
is a nodal circle, whose radius is given by the lowest root of J x (kr) = ; this 
makes r— '7l9af. 

* See Eayleigh's treatise, Art. 339. A general formula for calculating the roots of J 8 ' (ka) = Q, 
due to Prof. J. M c Mahon, is given in the special treatises. 

t The oscillations of a liquid in a circular basin of any uniform depth were discussed by 
Poisson, " Sur les petites oscillations de l'eau contenue dans un cylindre," Ann. de Gergonne, 
xix. 225 (1828-9) ; the theory of Bessel's Functions had not at that date been worked out, and 
the results were consequently not interpreted. The full solution of the problem, with numerical 
details, was given independently by Eayleigh, Phil. Mag. (5), i. 257 (1876) [Papers, i. 25]. 

The investigation in the text is limited, of course, to the case of a depth small in comparison 
with the radius a. Poisson's and Eayleigh's solution for the case of finite depth will be noticed 
in Chapter ix. 



288 



Tidal Waves 



[chap. VIII 




i9i] Properties of BesseVs Functions 289 

A comparison of the preceding investigation with the general theory of small oscilla- 
tions referred to in Art. 168 leads to several important properties of Bessel's Functions. 

In the first place, since the total mass of water is unaltered, we must have 



/2ir fa 
o J 



{rdOdr = 0, (16) 



where £ has any one of the forms given by (4). For s > this is satisfied in virtue of the 
trigonometrical factor cos s6 or sin s6 ; in the symmetrical case it gives 



/ 



J (kr)rdr=0 (17) 

o 



Again, since the most general free motion of the system can be obtained by super- 
position of the normal modes, each with an arbitrary amplitude and epoch, it follows 
that any value whatever of £, which is subject to the condition (16), can be expanded in a 
series of the form 

(= 22 (A, cos s6 + B 8 sin s6)J 8 (kr), (18) 

where the summations embrace all integral values of s (including 0) and, for each value of 
5, all the roots k of (5). If the coefficients A 8 , B 8 be regarded as functions of t, the equa- 
tion (18) may be regarded as giving the value of the surface-elevation at any instant. The 
quantities A 8 , B 8 are then the normal co-ordinates of the present system (Art. 168) ; and in 
terms of them the formulae for the kinetic and potential energies must reduce to sums of 
squares. Taking, for example, the potential energy 

v=isrp!iC*d*<ty, (19) 



/2tt ra 
o jo 



this requires that / / w 1 w 2 rd6dr=0, (20) 



where w u w 2 are any two terms of the expansion (18). If w it w 2 involve cosines or sines of 
different multiples of 0, this is verified at once by integration with respect to 6 ; but if 
we take 

w x oc J 8 (k x r) cos s0, w 2 « e/g far) cos s6, 

where k x , k 2 are any two distinct roots of (5), we get 

J 8 (k x r)J 8 (k 2 r)rdr = (21) 



/; 



o 
The general results, of which (17) and (21) are particular cases, are 



/: 



'J (kr)rdr= -^J '(ka) (22) 

(cf. Art. 102 (10)), and 



/ J* far) J 8 far) rdr= _ {k 2 aJ 8 ' faa) J 8 fad) - k x aJ 8 ' fad) J 8 fad)}. ...(23) 

In the case of k x = k 2 the latter expression becomes indeterminate; the evaluation in the 
usual manner gives 



/ 



a {Js(^)Yrdr = ~[Pa^{J 8 '(ka)^-h(k^-s^{J 8 (ka)Y] (24) 



For the analytical proofs of these formulae we refer to the treatises cited on p. 136. 

The small oscillations of an annular sheet of water bounded by concentric 
circles are easily treated, theoretically, with the help of Bessel's Functions of 
the second kind/ The only case of any special interest, however, is when the 
two radii are nearly equal ; we then have practically a re-entrant canal, and 
the solution follows more simply by the method of Art. 178. 



290 Tidal Waves [chap, viii 

The analysis can also be applied to the case of a circular sector of any 
angle*, or to a sheet of water bounded by two concentric circular arcs and 
two radii. 

An approximation to the frequency of the slowest mode in an elliptic basin of uniform 
depth can be obtained by Rayleigh's method, referred to in Art. 168. 



The equation of the boundary being 



a. + fc" 1 -* (25) 



let us assume, for the component displacements, 

(26) 



*-H 



SM>. 



n a 2 ' 



where the constants have been adjusted so as to make 

3*?-* <w 

at the boundary (25). The time-factor cos at is understood. The corresponding surface- 
elevation is 



f~»(g+g)-s<"+*>* 



.(28) 



The assumption (26) is however too general for the present purpose, since it includes 
circulatory motions. The condition of zero vorticity requires 

(2a 2 + b 2 )B=2a 2 A (29) 

We find from (26) 

2T=phj[(£ 2 + r} 2 ) dxdy = 2npabho- 2 Ua 2 + t \AB + h-^ + ^^j B*l sin 2 at, ...(30) 

2V=gp (U 2 dxdy=2nabgh 2 .2^^ cos 2 at (31) 

Expressing that the mean value of T — V is zero, and introducing the relation (29), we find 

„ 18a 2 + 6& 2 c 2 



5a 2 + 26 2 'a 2 ' 



■(32) 



where c 2 =gh. 

If we put b = a, this makes aa/c = 1*852, the true value for the circular basin being 
1*841. The approximate estimate is in excess, in accordance with a general principle 
(Art. 168). The various modes of longitudinal oscillations in an elliptic canal have been 
studied by Jeffreys "j* and Goldstein J, and more recently by Hidaka§, by different methods. 
It appears that in the gravest mode o-ajc = 1 '8866, whilst if we make bja-^0 in (32) we get 
a-a/c = 1 '8994. It would appear that the formula gives a good approximation for values of 
bja less than unity. 

* See Rayleigh, Theory of Sound, Art. 339. 
f Proc. Lond. Math. Soc. (2) xxiii. 455 (1924). 
+ Ibid, xxviii. 91 (1927). 

§ Mem. Imp. Mar. Obs. (Japan), iv. 99 (1931). This paper includes the discussion of the free 
oscillations in basins with boundaries of various other shapes, and with various laws of depth. 



191-193] Basin of Variable Depth 291 

192. As an example of forced oscillations in a circular basin, let us suppose 
that the disturbing forces are such that the equilibrium elevation would be 



?=<) 



cos s6 . cos (at + e) (33) 

This makes ^^=0, so that the equation (12) of Art. 189 reduces to the 

form (1), above, and the solution is 

%= A J s (kr) cos sd. cos (at + e), (34) 

where A is an arbitrary constant. The boundary-condition (Art. 189 (14)) 

gives 

AkaJg (ka) = sG, 

q T ( ]cf\ 

whence ? = G , //,» \ cos s6 . cos (at + e) (35) 

The case s = 1 is interesting as corresponding to a uniform horizontal 
force ; and the result may be compared with that of Art. 179. 

From the case s — 2 we could obtain a rough representation of the semi- 
diurnal tide in a polar basin bounded by a small circle of latitude, except that 
the rotation of the earth is not as yet taken into account. 

We notice that the expression for the amplitude of oscillation becomes 
infinite when J/ (ka) = 0. This is in accordance with a general principle, of 
which we have already had several examples ; the period of the disturbing 
force being now equal to that of one of the free modes investigated in the 
preceding Art. 

193*. When the sheet of water is of variable depth, the calculation at 
the beginning of Art. 189 gives, as the equation of continuity, 

9r_ d(hu) d(hv) 

dt dx dy K } 

The dynamical equations (Art. 189 (2)) are of course unaltered. Hence, 
eliminating f, we find, for the free oscillations, 

dt 2 
If the time-factor be e i(<Tt+ *\ we obtain 



-»&(»£K(*©I < 2 > 



£(»S+4«K<" < 3 > 



dx 

When h is a function of r, the distance from the origin, only, this may be 
written 

^+fl#=° w 

As a simple example we may take the case of a circular basin which shelves gradually 
from the centre to the edge, according to the law 



A = Ao ( 1 ~5) (5) 



* This formed Art. 189 of the 2nd ed. of this work (1895). A similar investigation was given 
by Poincar^, Legons de mecanique celeste, iii. 94 (Paris, 1910). 



292 Tidal Waves [chap, viii 

Introducing polar co-ordinates, and assuming that £ varies as cos s6 or sin s6. the equation 
(4) takes the form 



\ l a 2 )\dr 2 + rdr r 2 V a* r dr + ghj °* 



(6) 



That integral of this equation which is finite at the origin is easily found in the form 
of an ascending series. Thus, assuming 



t=2A, 



y > & 



where the trigonometrical factors are omitted, for shortness, the relation between consecu- 
tive coefficients is found to be 

(m 2 -s 2 )A m = \m(m-2)-s 2 -^-i A m _ 2 , 

a 2 a 2 
or, if we write — j— =n (n— 2)-s 2 , (8) 

where n is not as yet assumed to be integral, 

(m 2 -s 2 ) A m =(m-n) (m + n-Z) A m _ 2 ( 9 ) 

The equation is therefore satisfied by a series of the form (7), beginning with the term 
A 8 (rja) 8 , the succeeding coefficients being determined by putting m = s + 2, s+4, ... in (9). 
We thus find 

t- A f-Yll - ( n - s -2)( n + s ) ^ , (n-s-4) (n-s-2) (n + s) (n + s + 2) r*__ \ 
Q ~ 8 \a)\ 2(2s + 2) a 2+ 2. 4 (2s + 2) (2s + 4) a 4 "'J ' { } 

or in the usual notation of hypergeometric series 

{=A.?.F(a,f},y,£) (11) 

where a=i»i+Js, ^ = l+^s-^n, y=s + l. 

Since these make -y-a — /3 = 0, the series is not convergent for r=a, unless it terminate. 
This can only happen when n is integral, of the form s ■+- 2j. The corresponding values of 
o- are then given by (8). 

In the symmetrical modes (s = 0) we have 

C= A <>Y I2~^ + W72 2 ^"-J' {l2) 

where j may be any integer greater than unity*. It may be shewn that this expression 
vanishes for^' — 1 values of r between and a, indicating the existence of j— 1 nodal circles. 
The value of o- is given by 

o*-VU"l)*pP. 03) 

whence o-a/ N /(^ ) = 2 * 828 5 4 ' 899 > 6 ' 928 > ( 13a ) 

The gravest symmetrical mode (j=-2) has a nodal circle of radius '707 a. 

Of the unsymmetrical modes, the slowest, for any given value of 5, is that for which 
7i=s + 2, in which case we have 

£= A 8 — cos s0 cos (a-t + e). 



a» 



the value of o- being given by <r 2 =2s ,~ (14) 



9 k o 



In the case s=l the various frequencies are given by 



^=(4/ 2 -2)4°, (is; 



a' 



whence aa/Jfaho) = 1-414, 3'742, 5-831, (16) 

* If we put r/a.= sin £x> the series is identical with the expansion of -Py-i (cos x) 5 see Art. 85 (4). 



193-194] BesseTs Function of the Second Kind 293 

In the slowest of these modes, corresponding to s=l, w=3, the free surface is always 
plane. It appears from Art. 191 (15 a) that the frequency is '768 of that of the corre- 
sponding mode in a circular basin of uniform depth A , and of the same radius* 

As in Art. 192 we could at once write down the formula for the tidal motion produced 
by a uniform horizontal periodic force ; or, more generally, for the case where the disturbing 
potential is of the type 

Q, oc r s COS s6 COS (at + e). 

194. We may conclude this discussion of 'long' waves on plane sheets of 
water by an examination of the mode of propagation of disturbances from 
a centre in an unlimited sheet of uniform depth. For simplicity, we will 
consider only the case of symmetry, where the elevation £ is a function of 
the distance r from the origin of disturbance. This will introduce us to some 
peculiar and rather important features which attend wave-propagation in two 
dimensions. 

The investigation of a periodic disturbance involves the use of a Bessel's 
Function (of zero order) 'of the second kind,' as to which some preliminary 
notes may be useful. 

To solve the equation -=-£ + - -^- + <£ = (1) 



by definite integrals, we assume f cf>= l e~ zt Tdt, 



•(2) 



where T is a function of the complex variable t, and the limits of integration are constants 
as yet unspecified. This makes 
dty d<j> 



Z dz* 



by a partial integration. The equation (1) is accordingly satisfied by 

t-jwik' (3) 

provided the expression J{ 1 + 1 2 ) e ~ zt 

vanishes at each limit of integration. Hence, on the supposition that z is real and positive, 
or at all events has its real part positive, the integral in (3) may be taken along a path 
joining any two of the points t, —i, +oo in the plane of the variable t ; but two distinct 
paths joining the same points will not necessarily give the same result if they include 
between them one of the branch -points (t= ±i) of the function under the integral sign. 

Thus, for example, we have the solution 

e-^dt 



*-/-, 



N/(i+* 2 r 

where the path is the portion of the imaginary axis which lies between the limits, and that 
value of the radical is taken which becomes =1 for t=0. If we write t = g + ir), we obtain 

/l e~ izr >dr} n*" 
—j— ^ =2i I cos (zcosS) d$ = iirJ (z), (4) 
-W(l-»7 2 ) Jo 

which is the solution already met with (Art. 1 00). 

* For the oscillations in an elliptic basin with a similar law of depth see Goldsbrough, Proc. 
Roy. Soc. A, cxxx. 157 (1930). 

f Forsyth, Differential Equations, c. vii. The systematic application of this method to the 
theory of Bessel's Functions is due to Hankel, "Die Cylinderfunktionen erster u. zweiter Art," 
Math. Ann. i. 467 (1869). 



294 Tidal Waves [chap, viii 

An independent solution is obtained if we take the integral (3) along the axis of rj from 
the point (0, i) to the origin, and thence along the axis of £ to the point (oo , 0). This 
gives, with the same determination of the radical, 



d{irj) Z* 00 e~^d^ _ f 00 e~*d% . /"I e-***d v 



fO e -i*Vd(iri) .("> e -*d£; _ f 



,(5) 



By adopting other pairs of limits, and other paths, we can obtain other forms of <j), but 
these must all be equivalent to fa or fa, or to linear combinations of these. In particular, 
some other forms of fa are important. It is known that the value of the integral (3) taken 
round any closed contour which excludes the branch- 
points (t = ± i) is zero. Let us first take as our contour 
a rectangle, two of whose sides coincide with the positive 
portions of the axes of | and rj, except for a small semi- 
circular indentation about the point t = i, whilst the 
remaining sides are at infinity. It is easily seen that 
the parts of the integral due to the infinitely distant 
sides will vanish, either through the vanishing of the i ^)— 
factor e~ z % when £ is infinite, or through the infinitely 
rapid fluctuation of the function e~ iz1 >jr} when n is in- 
finite. Hence for the path which gave us (5) we may 
substitute that which extends along the axis of n from 
the point (0, i) to (0, i oo ), provided the continuity of 
the radical be attended to. Now as the variable t 

travels counter-clockwise round the small semicircle, the radical changes continuously 
from J ( 1 - ?7 2 ) to i J (rj 2 — 1 ). We have therefore 



.(6) 



*■-/< H/(^ 2 -l)"ii sW-irJo 6 dU 

It will appear that this solution is the one which is specially appropriate to the case of 
diverging waves. Another method of obtaining it will be given in Chapter x. 

If we equate the imaginary parts of (5) and (6) we obtain 

2 f 00 
Jq{ z )~ -\ sm ( z cosa u ) du, (7) 

7TJ o 

a form due to Mehler*. 

On account of the physical importance of the solution (6) it is convenient to have a 
special notation for it. We write f 

D (z) = - \ e - iecoshu du (8) 

"■y o 

This is equivalent to D (z)= — Y (z) — iJ (z), (9) 

where J F (z) = — — \ cos (z cosh u) du (10) 

IT J 

Equating the real parts of (5) and (6) we have, also, 

2 f 00 2 f$ n 

Y«(z)=--\ e-* sinhu du + - sin (z cos S) dS (11) 

tr J o "J 

* Math. Ann. v. (1872). 

f The use of a simple notation to meet the case of diverging waves seems justifiable. Our 
D (z) is equivalent to - iH W (z) in Nielsen's notation, as slightly modified by Watson. 

J This is the notation definitely recommended by Watson. The reader should be warned, 
however, that the same symbol has been employed by other writers in various senses. From 
a purely mathematical point of view the choice of a standard solution 'of the second kind' is 
largely a matter of convention, since the differential equation (1) is still satisfied if we add any 
constant multiple of J (z). Tables of the function Y (z) as defined by (10) are given in Watson's 
treatise. 



194] Asymptotic Expansion 295 

For a like reason, the path adopted for <j> 2 may be replaced by the line drawn from the 
point (0, i) parallel to the axis of £ (viz. the dotted line in the figure). To secure the con- 
tinuity of jj(l + t 2 ), we note that as t describes the lower quadrant of the small semicircle, 
the value of the radical changes from J (I -rf-) to e* ln V(2£), approximately. Hence along 
the dotted line we have, putting *=i+£, 

V(l+0=e*"V(2£-^ 2 ), 

where that value of the radical is to be chosen which is real and positive when £ is infini- 
tesimal. Thus 



(12) 



p +< ^ rf tf + Q 1 _^ g) f^ ^ 

V2 Ji **V(2£-^ 2 ) ^ 2 io $ K igj g 

If wc expa^u Jie binomial, and integrate term by term, we find 

™-®'<™W@+ 1J £(ff+-~}> w 

where use has been made of the formulae 



z* z* 



(14) 

.-ztrm-kj? H (*~B _ 1 ■ 3 ... (2m- 1) tt* 

If we separate the real and imaginary parts of (13) we have, on comparison with (9), 
Mz)=(^- Z ) {Bsm(z + i7r)-Scos(z + %7r)}, (15) 



Fo( * )= ~Gl) { EG0S ( z +fr)+ S *™( z +i«)}, 



.(16) 



1 2 .3 2 1 2 .3 2 .5 2 .7 2 ^ 

where R =i -__ + __^ _,.,, ^ 

l 2 1 2 .3 2 .5 2 I (1 ' 

ll(8i) 3!(82) 3+- -- J 

The series in (13) and (17) are of the kind known as 'semi-convergent,' or 'asymptotic,' 
expansions ; i.e. although for sufficiently large values of z the successive terms may for a 
while diminish, they ultimately increase again indefinitely, but if we stop at a small term 
we get an approximately correct result*. This may be established by an examination of 
the remainder after m terms in the process of evaluation of (12). 

It follows from (15) that the large roots of the equation J (z) = approximate to those of 

sin(z + J;7r) = (18) 

The series in (13) gives ample information as to the demeanour of the function D (z) 
when z is large. When z is small, D Q (z) is very great, as appears from (8). An approxi- 
mate formula for this case can be obtained as follows. Keferring to (11), we have 

/;^-*.^--(-i)*-/;^{x +5£+ i^ + ...}* 

=/;^>^(S-> ™ 

* Cf. Whitfcaker and Watson, Modern Analysis, c. viii.; Bromwich, Theory of Infinite Series, 
London, 1908, c. xi. ; Watson, c. vii. ; Gray and Mathews, c. iv. The semi-convergent 
expansion of J (z) is due to Poisson, Journ. de VEcole Polyt. cah. 19, p. 349 (1823); a rigorous 
investigation of this and other analogous expansions was given by Stokes, I.e. ante p. 285. The 
'remainder' was examined by Lipschitz, Crelle, lvi. 189 (1859). Cf. Hankel, I.e. ante p. 293. 



296 Tidal Waves [chap, viit 

The first term gives* 



/QO Q-W 
dw=-y-\og\z+., 



.(20) 



and the remaining ones are small in comparison. Hence, by (9) and (11), 

D (s)=--(log^ + y+^7r + ...) (21) 

IT 

It follows that 

lim«Z> '(«)=--t (22) 

The formula (21) is sufficient for our purposes, but the complete expression can now be 
obtained by comparison with the general solution of (1) in terms of ascending series, viz. J 



f Z 2 2 4 2 6 1 

<t> = AJ (z) + B y (z)\ogz + - 2 -s 2 ^-^ 2 + s 3 227427^2- •••[ 



.(23) 



where n m = l +^ + 3 + — + -• 

In order to identify this with (21), for small values of z, we must make 

B=--, 4=--(log^+ 7 +i*V) (24) 

7T 7T 

Hence 

2 2 (2 2 S 4 2 6 ) 

D (z)=--(\og fe+y+frV) J (z)~- |p-%22— p+ g 3 g2 ; 42 ; 62 " -J • - (25) 

195. We can now proceed to the wave-problem stated at the beginning 
of Art. 194. For definiteness we will imagine the disturbance to be caused 
by a variable pressure p applied to the surface. On this supposition the 
dynamical equations near the beginning of Art. 1 89 are replaced by 

du _ _ 3f _ 1 dp dv _ _ 1 9f _ 1 dp m 

dt dx p dx dt pdy p dy ' 

I=-4M) ( 2 > 

as before. 

If we introduce the velocity-potential in(l), we have, on integration, 

8-* + ? v 

We may suppose that p refers to the change of pressure, and that the arbi- 
trary function of t which has been incorporated in <£ is chosen so that d(j>/dt = 
in the regions not affected by the disturbance. Eliminating f by means of 
(2), we have 

S=^+;t <*> 

When (f> has been determined, the value of f is given by (3). 

* De Morgan, Differential and Integral Calculus, London, 1842, p. 653. 

t The Bessel's Functions of the second kind were first thoroughly investigated and made 
available for the solution of physical problems in an arithmetically intelligible form by Stokes, 
in a series of papers published in the Camb. Trans. With the help of the modern Theory of 
Functions, some of the processes have been simplified by Lipschitz and others, and (especially 
from the physical point of view) by Kayleigh. These later methods have been used in the text. 

I Forsyth, Differential Equations, c. vi. note 1; Watson, Bessel Functions, pp. 59, 60. 



194-195] Waves Diverging from a Centre 297 

We will now assume that p is sensible only over a small* area about the 
origin. If we multiply both sides of (4) by SocSy, and integrate over the area 
in question, the term on the left-hand side may be neglected (relatively), and 
we find 

d i ds= jphjt\[t* dxd y' (5) 

where 8s is an element of the boundary of the area, and 8n refers to the hori- 
zontal normal to 8s, drawn outwards. Hence the origin may be regarded as 
a two-dimensional source, of strength 

/«-*** (6) 

where P is the total disturbing force. 

Turning to polar co-ordinates, we have to satisfy 

g_,g*. + !g*), (7) 

or \3r 2 r dr/ 



where c 2 = gh, subject to the condition 



lim 



(_fcrg)-/ W (8) 



where f(t) is the strength of the source, as above defined. 

In the case of a simple-harmonic source e^ the equation (7) takes the 
form 

^1+^ = 0. W 

where k = a/c, and a solution is 

(t> = iD (kr)e^, (10) 

where the constant factor has been determined by Art. 194(22). Taking the 
real part we have 

(p=i {Jo (kr) sin at — Y (kr) cos at}, (11) 

corresponding to f (t) = cos at. 

For large values of kr the result (10) takes the form 

i iff [ t-- j -\iir 

*=v(8^r (12) 

The combination t — r/c indicates that we have, in fact, obtained the solution 
appropriate to the representation of diverging waves. 

It appears that the amplitude of the annular waves ultimately varies 
inversely as the square root of the distance from the origin. 

* That is, the dimensions of the area are small compared with the 'length' of the waves 
generated, this term being understood in the general sense of Art. 172. On the other hand, the 
dimensions must be supposed large in comparison with h. 



298 Tidal Waves [chap, viii 

196. The solution we have obtained for the case of a simple-harmonic 
source e* * may be written 

/•oo itrit- - cosh u ] 
2tt</)=| e V c y ck (13) 

This suggests generalization by Fourier's Theorem ; thus the formula 

27r<£=| fit — coshujdu (14) 

should represent the disturbance due to a source f(t) at the origin*. It is 
implied that the form of fit) must be such that the integral is convergent ; 
this condition will as a matter of course be fulfilled whenever the source has 
been in action only for a finite time. A more complete formula, embracing 
both converging and diverging waves, is 

2tt(/)= fU — ^coshujdu+j F (t 4- - cosh u J du (15) 

The solution (15) may be verified, subject to certain conditions, by substitution in the 
differential equation (7). Taking the first term alone, we find 

= / jsinh 2 u.f" (t — cosh u) — cosh u.f (t — cosh u H du 

=—,\ n/U — cosh u ) du= — sinh u . f [ t — cosh u ) 
r'Jo du* J \ c J r{_ J \ c J_\ u =o 

This obviously vanishes whenever f(t)=0 for negative values of t exceeding a certain 
limit f. 

Again — 27r?--^ = -/ coshw./'U — cosh u\du 

— - j (sinh u + e~ u )f U--cosh?tj du 

— -I / U~- cosh u\\ +-/ e~ u f (t--coshu \du 

=/ ( < -3 + "c/o 00e "" / ('"o- cosh K ) A 

under the same condition. The limiting value of this when r^-0 is f (t) ; and the state- 
ment made above as to the strength of the source in (14) is accordingly verified. 

A similar process will apply to the second term of (15) provided F (t) vanishes for 
positive values of t exceeding a certain limit. 

197. We may apply (14) to trace the effect of a temporary source varying 
according to some simple prescribed law. 

If we suppose that everything is quiescent until the instant t — 0, so that 

* The substance of Arts. 196, 197 is adapted from a paper "On Wave -Propagation in Two 
Dimensions," Proc. Lond. Math. Soc. (1), xxxv. 141 (1902). A result equivalent to (14) was obtained 
(in a different manner) by Levi-Civita, Nuovo Cimento (4), vi. (1897). 

t The verification is very similar to that given by Levi-Civita. 



196-197] Waves Diverging from a Centre 299 

f{t) vanishes for negative values of t, we see from (14) or from the equivalent 
form 

2 ^ri ag (i6) 

that </> will be zero everywhere so long as t < r/c. If, moreover, the source 
acts only for a finite time r, so that f(t) = for t > t, we have, for t > r + r/c, 



rr /(fl)rffl (17) . 



This expression does not as a rule vanish ; the wave accordingly is not sharply 
defined in the rear, as it is in front, but has, on the contrary, a sort of 'tail'f 
whose form, when t — r/c is large compared with t, is determined by 

2 ^=^T7^/ T /(*><** < 18 > 

The elevation f at any point is given by (3), viz. 

*-W (19) 

It follows that 



/: 



£eft=0, (20) 



provided the initial and final values of cf> vanish. It may be shewn that this 
will be the case when/(£) is finite and the integral 



f • 

J -00 



/(*)* (21) 



is convergent. The meaning of these conditions appears from (6). It follows 
that even when dP /dt is always positive, so that the flux of liquid in the 
neighbourhood of the origin is altogether outwards, the wave which passes 
any point does not consist solely of an elevation (as it would in the corre- 
sponding one-dimensional problem) but, in the simplest case, of an elevation 
followed by a depression. 

To trace in detail the progress of a solitary wave in a particular case we may assume 

/»-*£?■ (22 > 

which makes P increase from one constant value to another according to the law 

^=.4+5 tan- 1 - (23) 

* Analytically, it may be noticed that the equation (4), when_p = 0, may be written 

gy ay ay _ n 

daP^dfp^diict?' 

and that (17) consists of an aggregate of solutions of the known type 

{xt + yz + iictf}-*. 
t The existence of the ' tail ' in the case of cylindrical electric waves was noted by Heaviside, 
Phil. Mag. (5), xxvi. (1888) [Electrical Papers, ii.]. 



300 



Tidal Waves 



[chap, viii 



The disturbing pressure has now no definite epoch of beginning or ending, but the range 
of time within which it is sensible can be made as small as we please by diminishing r. 
For purposes of calculation it is convenient to assume 



/«-,- 



..(24) 



in place of (22), and to retain in the end only the imaginary part. We have then 

dz 
2tt(£ = 



f 00 du fi dz 

I t-- coshu-iT I t ir-(t-\ it) z 2 

J o c J o c \ c J 



.(25) 



where z = tanh hi. We now write 



Z-ir^cfo-v* t -r--iT=b 2 e- 2 ^, 



.(26) 



where we may suppose that a, b are positive, and that the angles a, /3 lie between and ^n. 
Since 



--H 



+ r 2, b * = { t + L)+ T 2 



tan 2a = , tan 28 

ct — r 



.(27) 



ct + r' 



it appears that a < b according as t^ 0, and that a > /3 always. With this notation, we find 



2ti-0 = 2 I 1 - 






2 e ~2ia_f ) 2 e -2ip z 2 



ab 



.(28) 



z " e -i(a-d) 




To interpret the logarithms, let us mark, in the plane of a complex variable z, the points 

o 

Since the integral in the second member of (28) is to be taken along the path 01, the proper 
value of the third member is 
<*(« + £) 



ab 



{(i* g+f. op/)- (kg |$-<. <*/)}, 




where real logarithms and positive values of the angles are to be understood. Hence, 
rejecting all but the imaginary part, we find 



«*-*i£ffli*S + =§fcs<.-«» 



.(29) 



as the solution corresponding to a source of the type (22). Here 

IP _ ( a 2 + 2ab cos (a- fl) + 6 2 \£ 2a6sin(a-/3) 

IQ V-2a6cos(a-/3) + W ' tan/ ^" 52^2 

and the values of a, b, a, |8 in terms of r and £ are to be found from (27). 



,(30) 



197-198] Solitary Wave 301 

It will be sufficient to trace the effect of the most important part of the wave as it 
passes a point whose distance r from the origin is large compared with cr. If we confine 
ourselves to times at which t - rjc is small compared with rjc, a. will be small compared 
with b, PIQ will be a small angle, and IPjIQ will = 1, nearly. If we put 

T 

t=- + T tan?;, (31) 

we shall have 

*=i*-h> a=V(rsec79), j8=icr/r, 6-(2r/c)*, (32) 

approximately ; and the formula (29) will reduce to 

2 ^ = ^ C0Sa= ^W cos d 7r -h)>/( cos n) (33) 

! 




The elevation £ is then given by 

2 "^=^j(^r^(vf s ™^-M cosi '> < 34 > 

approximately. The diagram shews the relation between £ and t, as given by this 
formula*. 

198. We proceed to consider the case of a spherical sheet, or ocean, of 
water covering a solid globe. We will suppose for the present that the globe 
does not rotate, and we will also in the first instance neglect the mutual 
attraction of the particles of the water. The mathematical conditions of the 
question are then exactly the same as in the acoustical problem of the 
vibrations of spherical layers of air*f*. 

Let a be the radius of the globe, h the depth of the fluid; we assume 
that h is small compared with a, but not (as yet) that it is uniform. The 
position of any point on the sheet being specified by the angular co-ordinates 
0, <f), let u be the component velocity of the fluid at this point along the 
meridian, in the direction of 6 increasing, and v the component along the 
parallel of latitude, in the direction of <£ increasing. Also let f denote the 
elevation of the free surface above the undisturbed level. The horizontal 

* The points marked - 1, 0, + 1 correspond to the times rjc - r, rjc, r/c + r, respectively, 
f Discussed in Eayleigh's Theory of Sound, c. xviii. 



302 Tidal Waves [chap, viii 

motion being assumed, for the reasons explained in Art. 172, to be the same 
at all points in a vertical line, the condition of continuity is 

4 (uha sin 686) 86 + ^- (vha 86) 8cf> = - a sin 68<f> . a86 . |f , 
otf ocp 01 

where the left-hand side measures the flux out of the columnar space 

standing on the element of area a sin 68(f) . a 86, whilst the right-hand member 

expresses the rate of diminution of the volume of the contained fluid, owing 

to fall of the surface. Hence 

3? 1_ \ d(hu sin 6) d(hv)\ 

dt~ asin6\ d6 d<j> J W 

If we neglect terms of the second order in u, v, the dynamical equations 

are, on the same principles as in Arts. 169, 189, 

dt~ 9 add ad6' dt~ g asm6d<l> asm6d<t>' 

where fl denotes the potential of the extraneous forces. 

If we put Z = -n/g, (3) 

these may be written 

dt~ ad6 K * Ui dt asm6d<f>^ *' w 

Between (1) and (4) we can eliminate u, v, and so obtain an equation in f 
only. 

Iu the case of simple-harmonic motion, the time-factor being e i{<Tt+e) , the 
equations take the forms 

r _ % (d (hu sin 6/) d(hv)\ ,„. 

*~<rasm0( ~W " + ~df]' W 

-<&»«-& — "sifo^^ft (6) 

199. We will now consider more particularly the case of uniform depth. 
To find the free oscillations we put ? = ; the equations (5) and (6) of the 
preceding Art. then lead to 

^— - Q — n sintf^j + ^-o-5 5tI+ —r ? = (1) 

sin 6d6\ d6J sin 2 6d(f> 2 gh & v ' 

This is identical in form with the general equation of spherical surface- 
harmonics (Art. 83 (2)). Hence, if we put 



<7 2 a 2 



^r=^ +1 >' < 2 > 

a solution of (1) will be ? = S n , (3) 

where S n is the general surface-harmonic of order n. 

It was pointed out in Art. 86 that # n will not be finite over the whole 
sphere unless n be integral. Hence, for an ocean covering the whole globe, 



198-199] Waves on a Spherical Ocean 303 

the form of the free surface at any instant is, in any fundamental mode, that 
of a 'harmonic spheroid* 

r = a + h + S n cos (at + e), (4) 

and the speed of the oscillation is given by 

, -{»(» + l))i. <2*? (5) 

Co 

the value of n being integral. 

The characters of the various normal modes are best gathered from a 
study of the nodal lines (S n = 0) of the free surface. Thus, it is shewn in 
treatises on Spherical Harmonics * that the zonal harmonic P n (/jl) vanishes 
for n real and distinct values of /jl lying between + 1, so that in this case 
we have n nodal circles of latitude. When n is odd one of these coincides 
with the equator. In the case of the tesseral harmonic 

the second factor vanishes for n — s values of fi, and the trigonometrical 
factor for 2s equidistant values of <f>. The nodal lines therefore consist of 
n— s parallels of latitude and 2s meridians. Similarly the sectorial harmonic 

(l-^)& C0S \n<t> 
smj 

has as nodal lines 2n meridians. 

These are, however, merely special cases, for since there are 2n + 1 inde- 
pendent surface-harmonics of any integral order n, and since the frequency, 
determined by (5), is the same for each of these, there is a corresponding 
degree of indeterminateness in the normal modes, and in the configuration of 
the nodal lines. 

We can also, by superposition, build up various types of progressive 
waves ; e.g. taking a sectorial harmonic we get a solution in which 

f oc (1 - /a 2 )*** cos (w0 - <r* + €) ; (6) 

this gives a series of meridianal ridges and furrows travelling round the 
globe, the velocity of propagation, as measured at the equator, being 

aa /n+ 1\£ 



?-Pf •<** (7) 



It is easily verified, on examination, that the orbits of the particles are now 
ellipses having their principal axes in the directions of the meridians and 
parallels, respectively. At the equator these ellipses reduce to straight lines. 

In the case n = 1, the harmonic is always of the zonal type. The 
harmonic spheroid (4) is then, to our order of approximation, a sphere 
excentric to the globe. It is important to remark, however, that this case 

* For references see p. 110. 



304 Tidal Waves [chap, viii 

is, strictly speaking, not included in our dynamical investigation, unless we 
imagine a constraint applied to the globe to keep it at rest; for the de- 
formation in question of the free surface would involve a displacement of 
the centre of mass of the ocean, and a consequent reaction on the globe. 
A corrected theory for the case where the globe is free could easily be 
investigated, but the matter is hardly important, first because in such a 
case as that of the earth the inertia of the solid globe is so enormous 
compared with that of the ocean, and secondly because disturbing forces 
which can give rise to a deformation of the type in question do not as a 
rule present themselves in nature. It appears, for example, that the first 
term in the expression for the tide-generating potential of the sun or moon 
is a spherical harmonic of the second order (see the Appendix to this 
Chapter). 

When n = 2, the free surface at any instant is approximately ellipsoidal. 
The corresponding period, as found from (5), is then '816 of that belonging 
to the analogous mode in an equatorial canal (Art. 181). 

For large values of n the distance from one nodal line to another is 
small compared with the radius of the globe, and the oscillations then take 
place much as on a plane sheet of water. For example, the velocity of 
propagation, at the equator, of the sectorial waves represented by (6) tends 
with increasing n to the value (gh)%, in agreement with Art. 170. 

From a comparison of the foregoing investigation with the general theory of Art. 168 
we are led to infer, on physical grounds alone, the possibility of the expansion of any- 
arbitrary value of £ in a series of surface -harmonics, thus 

o 
the coefficients of the various independent harmonics being the normal co-ordinates of the 
system. Again, since the products of these coefficients must disappear from the expressions 
for the kinetic and potential energies, we are led to the ' conjugate ' properties of spherical 
harmonics quoted in Art. 87. The actual calculation of the energies will be given in the 
next Chapter, in connection with an independent treatment of the same problem. 

The effect of a simple-harmonic disturbing force can be written down at 
once from the formula (14) of Art. 168. If the surface value of O be 
expanded in the form 

n = xa n (8) 

where H w is a surface-harmonic of integral order n, the various terms are 
normal components of force, in the generalized sense of Art. 135 ; and the 
equilibrium value of f corresponding to any one term X2 B is 

Cn-.-Q.lg (9) 

Hence, for the forced oscillation due to this term, we have 

&— i-is^r 1 < 10) 

l—o- j<T n g 



i99-2oo] Free and Forced Oscillations 305 

where a measures the 'speed' of the disturbing force, and a n that of the 
corresponding free oscillation, as given by (5). There is no difficulty, of course, 
in deducing (10) directly from the equations of the preceding Art. 

200. We have up to this point neglected the mutual attraction of the 
parts of the liquid. In the case of an ocean covering the globe, and with 
such relations of density as we meet with in the actual earth and ocean, this 
is not insensible. To investigate its effect in the case of the free oscillations, 
we have only to substitute for X2 n , in the last formula, the gravitation - 
potential of the displaced water. If the density of this be denoted by p, 
whilst p represents the mean density of the globe and liquid combined, we 
have* 

_ jiTTvpa 

n ~ 2n + l U ' (11) 

and g =%y7rap , (12) 

7 denoting the gravitation-constant, whence 

fl »=-2irh>v^ < 13 > 

Substituting in (10) we find 

2H-2^)> ^ 

where cr n is now used to denote the actual speed of the oscillation, and <r n ' 
the speed calculated on the former hypothesis of no mutual attraction. 
Hence the corrected speed is given by 

3 p\gh 

2n + 1 p 0/ 

For an ellipsoidal oscillation {n = 2), and for p/p = '18 (as in the case of 
the Earth), we find from (14) that the effect of the mutual attraction is to 
lower the frequency in the ratio of *94 to 1. 

The slowest oscillation would correspond to n = 1, but, as already indicated, 
it would be necessary, in this mode, to imagine a constraint applied to the 
globe to keep it at rest. This being assumed, it appears from (15) that if 
p> po the value of a x 2 is negative. The circular function of t is then replaced 
by real exponentials; this shews that the configuration in which the surface 
of the sea is a sphere concentric with the globe is one of unstable equilibrium. 
Since the effect of the constraint is merely to increase the inertia of the 
system, we infer that the equilibrium is still unstable when the globe is free. 
In the extreme case where the globe itself is supposed to have no gravitative 

* See, for example, Eouth, Analytical Statics, 2nd ed., Cambridge, 1902, ii. 146-7. 

t This result was given by Laplace, Mecanique Celeste, Livre l er , Art. 1 (1799). The free and 
the forced oscillations of the type n = 2 had been previously investigated in his "Eecherches sur 
quelques points du systeme du monde," Mem. de V Acad. roy. des Sciences, 1775 [1778] [Oeuvres 
Completes, ix. 109, ...]. 



'- i -(-+i)(i-5^i3S < l5 >t 



306 Tidal Waves [chap, viii 

power at all, it is obvious that the water, if disturbed, would tend ultimately, 
under the influence of dissipative forces, to collect itself into a spherical mass, 
the nucleus being expelled. 

It is obvious from Art. 168, or it may easily be verified independently, 
that the forced vibrations due to a given periodic disturbing force, when the 
gravitation of the water is taken into account, will be given by the formula 
(10), provided Xl n now denote the potential of the extraneous forces only, and 
cr n have the value given by (15). 

201. The oscillations of a sea bounded by meridians, or parallels of 
latitude, or both, can also be treated by the same method*. The spherical 
harmonics involved are however, as a rule, no longer of integral order, and it 
is accordingly difficult to deduce numerical results. 

In the case of a zonal sea bounded by two parallels of latitude, we assume 

t={Ap(ri + Bq(ri} C ™\scl>, (1) 



whence 



.(3) 



sin, 

where /t=cos 6, and p (/x), q (it) are the two functions of it, containing (1 - it 2 )£ 8 as a factor, 
which are given by the formula (2) of Art. 86. It will be noticed that p (it) is an even, and 
q (fi) an odd function of p. 

If we distinguish the limiting parallels by suffixes, the boundary conditions are that 
u=0 for fi=fii and n = fi2' For the free oscillations this gives, by Art. 198 (6), 

Ap'M + Bq'fa^O, Ap'(n 2 ) + Bq'(fji 2 ) = 0, (2) 

p' M, 9.' M 
p' M, q' M 

which is the equation to determine the admissible values of n, the order of the harmonics- 
The speeds (<r) corresponding to the various roots are given as before by Art. 199 (5). 

If the two boundaries are equidistant from the equator, we have /x 2 = — /*i", The above 
solutions then break up into two groups ; viz. for one of these we have 

b=o, yw=o, (4) 

and for the other .4=0, q' (i^) = (5) 

In the former case £ has the same value at two points symmetrically situated on opposite 
sides of the equator ; in the latter the values at these points are numerically equal, but 
opposite in sign. 

If we imagine one of the boundaries to be contracted to a point (say /x 2 = 1), we pass to 
the case of a circular basin. The values of p' (1) and q' (1) are infinite, but their ratio can 
be evaluated by means of formulae given in Art. 84. This gives, by the second of equations 
(2), the ratio A : B, and substituting in the first we get the equation to determine n. 
A simpler method of treating this case consists, however, in starting with a solution 
which is known to be finite, whatever the value of n, at the pole it = l. This involves 
a change of variable, as to which there is some latitude of choice. We might take, for 
instance, the expression for P n 8 (cos 6) in Art. 86 (6), and seek to determine n from the 
condition that 

^P n »(cos0) = O (6) 

for = 6^. By making the radius of the sphere infinite, we can pass to the plane problem 
of Art. 191 J. The steps of the transition will be understood from Art. 100. 

* Cf. Rayleigh, I.e. ante p. 301. 

+ This question has been discussed by Macdonald, Proc. Lond. Math. Soc. xxxi. 264 (1899^ 

J Cf. Rayleigh, Theory of Sound, Arts. 336, 338. 



200-203] Waves on a Limited Ocean 307 

If the sheet of water considered have as boundaries two meridians (with or without 
parallels of latitude), say = and = o, the condition that v=0 at these restricts us to 
the factor cossco, and gives sa = mir, where m is integral. This determines the admissible 
values of s, which are not in general integral*. The diurnal and semi-diurnal tides in a 
non-rotating ocean of uniform depth bounded by two meridians have been studied by 
Proudman and Doodson, and worked out for special cases and for special depths f. 

Dynamics of a Rotating System. 

202. The theory of the tides on an open sheet of water is seriously 
complicated by the fact of the earth's rotation. If, indeed, we could assume 
that the periods of the free oscillations, and of the disturbing forces, were 
small compared with a day, the preceding investigations would apply as 
a first approximation, but these conditions are far from being fulfilled in the 
actual circumstances of the earth. 

The difficulties which arise when we attempt to take the rotation into 
account have their origin in this, that a particle having a motion in latitude 
tends to keep its angular momentum about the earth's axis unchanged, and 
so to alter its motion in longitude. This point is of course familiar in 
connection with Hadley's theory of the trade-winds {. Its bearing on tidal 
theory seems to have been first recognized by Maclaurin§. 

Owing to the enormous inertia of the solid body of the earth compared 
with that of the ocean, the effect of tidal reactions in producing periodic 
changes of the angular velocity is quite insensible. This angular velocity 
will therefore for the present be treated as constant. 

The theory of the small oscillations of a dynamical system about a state 
of equilibrium relative to a real or ideal rigid frame which rotates with con- 
stant angular velocity about a fixed axis differs in some important particulars 
from the theory of small oscillations about a state of absolute equilibrium, 
of which some account was given in ilrt. 168. It is therefore worth while to 
devote a little space to it before entering on the consideration of special 
problems. The system considered may be entirely free, or it may be connected 
with a rotating solid. In the latter case it is assumed that the connecting 
forces as well as the internal forces of the system are subject to the 'con- 
servative' law. 

203. The equations of motion of a particle m relative to rectangular 
axes Ox, Oy, Oz which rotate about Oz with angular velocity co are 

m (x - 2coy — a> 2 x) — X, m(y+2'*)x—<o 2 y)=Y, mz=Z, ...(1) 
where X, Y, Z are the impressed forces. 

* The reader who wishes to carry the study of the problem further in this direction is 
referred to Thomson and Tait, Natural Philosophy (2nd ed.), Appendix B, " Spherical Harmonic 
Analysis." 

f M. N. R. A. S., Geophy. Suppt. i. 468 (1927), and ii. 209 (1929). 

% "The Cause of the General Trade Winds," Phil. Trans. 1735. 

§ De Causd Physicd Fluxus et Refluxus Maris, Prop, vii.: "Motus aquee turbatur ex insequali 
velocitate qua corpora circa axem Terrse motu diurno deferuntur" (1740). 



308 Tidal Waves [chap, viii 

Let us now suppose that the relative co-ordinates {%, y, z) of each particle 
are expressed in terms of a certain number of independent quantities 
q 1} q 2 , ... q r . We write 

^jSm^ + ^ + i 2 ), T =|o) 2 2m(^ + 2/ 2 ) (2) 

Hence ® denotes the kinetic energy of the relative motion, which we 
shall suppose expressed as a homogeneous quadratic function of the generalized 
velocities q r , with coefficients which are functions of the generalized co- 
ordinates q r ; whilst T is the kinetic energy of the system when rotating, 
without relative motion, in the configuration (q lt q 2 , ... q n ). Finally we put 

2 (X&*+ YBy 4- ZBz) = -8V+ Q 1 8q 1 + Q 2 8q 2 + ... + Q n 8q ni ...(3) 

where Fis the potential energy and Q lt Q 2i ... Q n are the generalized com- 
ponents of extraneous force. 

If we multiply the three equations (1) by dx/dq r , dy/dq r) dz/dq r , respec- 
tively, and add, and sum the result for all the particles of the system, and 
then proceed as in the 'direct' proof of Lagrange's equations, we obtain the 
following typical equation of motion in generalized co-ordinates * : 

5f-£+At* + A-* + - + A-*'— 4 (F - W + *' - (4) 

n K Of 1/) 

where ft r8 = 2o>Sm - , ' % (5) 

It is to be noted that 

/3r S = -/3sr, /3 rr =0 (6) 

The equation (4) may also be derived from Art. 141 (23), with the help 
of Art. 142 (8), by supposing the rotating frame to be free, but to have an 
infinite moment of inertia. 

The conditions for relative equilibrium, in the absence of disturbing 
forces, are found by putting q 1} q 2y ... q r = in (4), whence 

a!< F - r °) = ' < 7 > 

shewing that the equilibrium value of V— T is 'stationary.' 
Again, from (1) we have 

Sw (xx + yy + zz) — a> 2 2m {xx + yy + zz) = 1 (Xx + Yy + Zz), . . .(8) 
or, by (2) and (3) 

j t ('&+V-To) = Q 1 q 1 + Q*qz+. ~ + Qnq n (9) 

This result may also be deduced from (4), taking account of the relations (6). 

* Cf. Thomson and Tait, Natural Philosophy (2nd ed.), i. 310; Lamb, Higher Mechanics , 
2nded., Art. 84. 



203-204] Dynamics of a Rotating System 309 

When there are no disturbing forces we have 

®+ F-r = const (10) 

The form assumed by the Hamiltonian theorem of Art. 135 is also to be 
noticed. The total kinetic energy of our system is 

T=$2m\(a;- coy) 2 + {y + coxf + i 2 } = ® + T + coM, (11) 

where M=Xin(xy — yx) (12) 

If there are no extraneous forces we have 



JV 



V)dt = 0, (13) 

subject to the usual terminal condition. Hence 

a[ (® + T a + ©if- V)dt=0, (14) 

with the condition 

Xm {(x — coy) Ax + (y + ax) Ay 4- zAz}\ =0 (15) 

Jto 

This theorem may also be deduced directly from (1) by the usual 
Hamiltonian procedure, and leads in turn t6 an independent proof of the 
equations (4), for the case of free motion. The inclusion of disturbing forces in 
the investigation presents no difficulty. 

The condition (15) is fulfilled whenever the initial and final relative con- 
figurations are the same in the varied as in the actual motion. 

204. We will now suppose the co-ordinates q r to be chosen so as to vanish 
in the undisturbed state. In the case of a small disturbance, we may then 
write 

2®< = a n q 1 2 + a 22 q<?+ ... +2a 12 q t q 2 + ... , (1) 

2(F-T )=c 11 gi 2 + c 22 g 2 2 -f-...+2c 12 9^ 2 + ..., (2) 

where the coefficients may be treated as constants. The terms of the 
first degree in V — T have been omitted, on account of the 'stationary' 
property. 

In order to simplify the equations as much as possible, we will further 
suppose that, by a linear transformation, each of these expressions is reduced, 
as in Art. 168, to a sum of squares; viz. 

2® = a 1 q 1 2 + o 2 q 2 2 + ... +a n g n 2 , (3) 

2(V-T )=c l q 1 * + c 2 q 2 *+...+c n q 7 ? (4) 

The quantities q 1} q 2 , ... q n may be called the 'principal co-ordinates' of the 
system, but we must be on our guard against assuming that the same 
simplicity of properties attaches to them as in the case of no rotation. The 
coefficients a x , a 2 , ... a n and c 1} c 2 , ... c n may be called the 'principal co- 
efficients' of inertia and of stability, respectively. The latter coefficients 



310 



Tidal Waves 



[chap, vm 



are the same as if we were to ignore the rotation, and to introduce fictitious 
'centrifugal' forces (ma> 2 x, mcoPy, 0) acting on each particle in the direction 
outwards from the axis. 

The equations (4) of the preceding Art. become, in the case of infinitely 
small motions, 

«1 qi +C X q X + #12 J2 + £l3<Z3 + • • • + $lnin = Ql, 

d 2 q 2 + C 2 <72 + #21<?1 + /3 2 3<?3 + . . . + fiznqn = $2, 



-G.J 



,(5) 



Ctn'qn + C n q n + Pnl4l + &*&& + £ TC 3<?3 + 

where the coefficients j3 rs may be regarded as constants. 

If we multiply these by q x , q 2 , ... J n in order and add, we find, taking 
account of the relation /3 rs — — fi sr , 
d 



dt 



(® + F- T ) = QKJ! + $2^2+ ... + Qntfn, 



.(6) 



as has already been proved without approximation. 

205. To investigate the free motions of the system, we put Q lf Q 2 , ... 
Q n = 0, in (5), and assume, in accordance with the usual method of treating 
linear equations, 

q 1 = A 1 e", q 2 = A 2 e kt , ... q n =A n e kt (7) 

Substituting, we find 

(Oi\*+Qi)ili +i8iiX4i+... + /3 ln \A n = 0, 

fi 21 \A 1 + (a ? \ 2 + c 2 ) A 2 + ... +/3 2n \A n = 0, 



/3 nX \A x + /5 n2 X^ 2 + • • • + KX 2 + o n ) A n = 0. 

Eliminating the ratios A X :A 2 : ... :A n , we get the equation 
tfiX 2 + Ci, p X2 \, ... f3 Xn \ 

fi 2i \, Cl 2 \ 2 + C 2 , ... /3 2 nX 



.(8) 



&l\ 



/3 n2 X, ... a n X 2 + c„ 



= 0, 



.(9) 



or, as we shall occasionally write it, for shortness, 

D(\)=0 (10) 

The determinant D (X) comes under the class called by Cayley ' skew- 
determinants,' in virtue of the relations (6) of Art. 203. If we reverse the 
sign of X, the rows and columns are simply interchanged, and the value of the 
determinant is therefore unaltered. Hence the equation (10) will involve 
only even powers of X, and the roots will be in pairs of the form 

X= + (p +i<r). 

In order that the configuration of relative equilibrium should be stable 
it is essential that the values of p should all be zero, for otherwise terms of 
the forms e ±pt cos at and e ±pe sin at would present themselves in the realized 



204-205] Condition for Stability 311 

expression for any co-ordinate q r . This would indicate the possibility of an 
oscillation of continually increasing amplitude. 

In the theory of absolute equilibrium, sketched in Art. 168, the necessary 
and sufficient condition of stability (in the above sense) was simply that the 
potential energy must be a minimum in the configuration of equilibrium. In 
the present case the conditions are more complicated*, but it is easily seen 
that if the expression for V— T be essentially positive, in other words if the 
coefficients Ci, c%, ... c n in (4) be all positive, the equilibrium must be stable. 
This follows at once from the equation 

V + (F- To) = const., (11) 

proved in Art. 203, which shews that under the present supposition neither 
® nor V— T can increase beyond a certain limit depending on the initial 
circumstances f. It will be observed that this argument does not involve 
the use of approximate equations. 

Hence stability is assured if V — T is a minimum in the configuration 
of relative equilibrium. But this condition is not essential, and there may 
even be stability (from the present point of view) with V — T a maximum, 
as will be shewn presently in the particular case of two degrees of freedom. 
It is to be remarked, however, that if the system be subject to dissipative forces, 
however slight, affecting the relative co-ordinates q ly q%, ... q n , the equi- 
librium will be permanently or 'secularly' stable only if V— T is a minimum. 
It is the characteristic of such forces that the work done by them on the 
system is always negative. Hence by (6) the expression Q£ + (V — T ) will, so 
long as there is any relative motion of the system, continually diminish, in 
the algebraical sense. Hence if the system be started from relative rest in a 
configuration such that V — T is negative, the above expression, and therefore 
a fortiori the part V - T , ,vill assume continually increasing negative values, 
which can only take place by the system deviating more and more from its 
equilibrium-configuration. 

This important distinction between 'ordinary' or kinetic, and secular' 
or practical stability was first pointed out by Thomson and TaitJ. It is to 
be observed that the above investigation presupposes a constant angular 
velocity (g>) maintained, if necessary, by a proper application of force to the 
rotating solid. When the solid is free, the condition of secular stability takes 
a somewhat different form, to be referred to later (Chapter XII.). In the 

* They have been investigated by Kouth, I.e. ante p. 195 ; see also his Advanced Rigid 
Dynamics, c. vi. 

t The argument was originally applied to the theory of oscillations about a configuration of 
absolute equilibrium (Art. 168) by Dirichlet, " Ueber die Stabilitat des Gleichgewichts, "• Crelle, 
xxxii. (1846) [Werke, Berlin, 1889-97, ii. 3]. An algebraic proof is indicated in Higher Mechanics, 
2nd ed., Art. 99. 

J Natural Philosophy (2nd ed.), Part i. p. 391. See also Poincar£, " Sur l'equilibre d'une 
masse fluide animee d'un mouvement de rotation," Acta Mathematica, vii. (1885), and op. cit. 
ante p. 146. Some simple mechanical illustrations are given in a paper "On Kinetic Stability," 
Proc. Roy. Soc. A, lxxx. 168 (1909), and in the author's Higher Mechanics, 2nd ed., p. 253. 



312 Tidal Waves [chap, vm 

practical applications we shall be concerned only with cases where V— T is 
a minimum, and the coefficients Ci, c 2 , ... c n in Art. 204 (4) accordingly positive. 
To examine the character of a free oscillation, in the case of stability, we 
remark that if X be any root of (10), the equations (8) give 

^=— 2 =...=^=a, (12) 

«i a 2 a n 

where a lt a^, ... a n are the minors of any row in the determinant D (X), and G 

is arbitrary. These minors will as a rule involve odd as well as even powers of 

X, and so assume unequal values for the two oppositely signed roots (+ X) of 

any pair. If we put X = ± iar, the corresponding values of a r will be of the 

forms fi r ± iv r , where fj, r , v r are real. Hence 

q r = G (fj, r + iv r ) e i<rt + G' (fi r - iv r ) e~ ift K 
If we put G = \K&\ G' = \Ke- ie , 

we get a solution of our equations in real form, involving two arbitrary 
constants K, e; thus 

q r = K {fi r cos (at + e) — v r sin (at + e)} (13) 

This formula expresses what may be called a ' natural mode ' of oscillation 

of the system. The number of such possible modes is of course equal to the 

number of pairs of roots of (9), i.e. to the number of degrees of freedom of 

the system. It is to be noticed, as an effect of the rotation, that the various 

co-ordinates are no longer in the same phase. 

If £, t], £ denote the component displacements of any particle from its equilibrium 
position, we have 



. dx dx 




dy , dy 




. dz dz 


dz 



.(14) 



Substituting from (13), we obtain a result of the form 

g = P. K cos (<rt + e) + P' . K sin (<rt + e), \ 

V = Q. K cos (<rt + e) + Q' .K sin (<rt + e), I (15) 

£=R . iTcos (<rt+€) + R' . Ksin (<rt+c), J 
where P, P', Q, Q\ R, R' are determinate functions of the mean position of the particle, 
involving also the value of <r, and therefore different for the different normal modes, 
but independent of the arbitrary constants K, e. These formulae represent an elliptic- 
harmonic motion of period 27r/<r, the directions 

i-l-i and L-JL-A. da) 

being those of two conjugate semi-diameters of the elliptic orbit, of lengths 

(P 2 +Q 2 + R 2 )^. K, and (P' 2 + Q' 2 + R' 2 )l . K, 
respectively. The positions and forms and relative dimensions of the elliptic orbits, as 
well as the relative phases of the particles in them, are accordingly in each natural mode 
determinate, the absolute dimensions and epochs being alone arbitrary. 



205-205 b] Free Oscillations 313 

205 a. When the angular velocity co is small the normal modes will as a rule 
differ only slightly from the case of no rotation, and expressions for the altered 
types and frequencies can then be found as follows*. Since the determinantal 
equation (9) of Art. 205 is unaltered when we reverse the signs of all the &'s, 
the frequencies will usually involve these quantities in the second order. Hence, 
considering for example the mode in which A x is finite, whilst A 2 ,A 3) ... A n 
are relatively small, and writing \ = i<r ly the rth equation of the system (8) 
gives, approximately, 

A 1 _ ffirlOl m x 

Al a r (<Ti*-<r*y KU) 

where a r 2 = c r ja r . Hence, substituting in the first equation, we get a corrected 
value of o-] 2 ; thus 

<V = Ml + S - H i (18) 



«1 l r aidr Ol* ~ 0> 2 )J 

But these approximations fail if any denominator in the bracket vanishes 
or is even small. This case arises when two or more of the normal modes in 
the absence of rotation have the same or nearly the same period. Suppose, for 
instance, that a x 2 and a 2 2 are nearly equal. We have then, from (8), with 
A = icr, 

(ci 2 - a 2 a^ Ai + i/3 12 crA 2 = 0, 



•(19) 
^21^1 + (c 2 - o 2 a 2 ) A 2 = 0, 

so that Ax and A 2 are comparable. Eliminating Ai/Ai, we have 

(^-^)(^-^) = §^^ (20) 

In the case of exact equality this gives 

* 2 -" 2=± v(Sr- (21) 

*- i - ± «^53' (22) 

approximately. The change of frequency due to the rotation is now proportional 
to co instead of w 2 . 

The values of A z , A i} ... A n in terms of A 1} A 2 are to be found from the 
remaining equations of the system (8), but would only affect the above con- 
clusion by terms involving co 2 . 

205 b. On account of the analytical difficulties which attend the deter- 
mination of the free modes of oscillation, especially in the case of continuous 
systems, it is natural to look for an approximate method of calculating the 
more important frequencies, analogous to that employed by Rayleigh in the 
case of non-rotating systems (Art. 168). 

* Rayleigh, Phil. Mag. (6) v. 293 (1903) {Papers, v. 89]. 



314 Tidal Waves [chap, viii 

For this purpose we may have recourse to the variational formula (14) of 
Art, 203. In the application to small oscillations it is convenient to express 
this in terms of the displacements (f, rj, f) of the particles from their positions 
of relative equilibrium. Writing oo + f, y + 77, z + f for x, y, z, where x Q , yo,z 
refer to the equilibrium position, we have 

A \ tl Mdt = A \ h M'dt+ r2m(ar ^-yoAf)T\ (1) 

Jt J t L Jk 

where M! = 2m(&- v $) (2) 

When the integrated terms in (1) are incorporated in the terminal con- 
dition (15) of Art. 203, the theorem becomes 

&[ tl (® + coM' + To-V)dt = O i (3) 

J t 

with the condition 



2m {($ - co V ) Af + (v + »|) At; + ?Af } 



= (4) 



Let us now suppose that the varied, as well as the natural motion, is 
simple-harmonic with the same period 2ir/a y and that the limits of integration 
£0, £1 differ by an exact period. The terms in (4) which relate to the two 
limits will then cancel, so that the postulated condition is fulfilled. The 
result is that the mean value (with respect to time) of the expression 

W+uM'-iV-To) (5) 

is stationary for small arbitrary variations of the type of vibration, the period 
being kept constant. 

In terms of generalized co-ordinates (assumed to vanish in relative equi- 
librium) M' will be a bilinear function of the two sets of variables 

qi,q2> ... q n and q lt q 2i ... q ny 
whilst ® and V—T are already by hypothesis homogeneous quadratic 
functions of the velocities and co-ordinates, respectively. Hence (5) is a 
homogeneous quadratic function of the variables q r , q r . 

If we now write 

q r = A r cos at + B r sin at, (6) 

and denote the resulting mean value of the expression (5) by J, we have 

J=a*P + aQ-R y (7) 

where P, Q, R are certain homogeneous quadratic functions of the variables 
A r , B ry whose precise forms are not required for the moment. 
The stationary property asserts that 

<7 2 AP + <tAQ-AE = (8) 

for all infinitesimal values of AA r ,Ai? r . In particular, putting A^l r = e^ r , 
&B r =eB r , where e is an infinitesimal constant independent of r, we have 

^=0, (9) 



205 b] Free Periods; Approximations 315 

on account of the homogeneous character. The statement that in a free 

oscillation the mean value of the expression (5) is zero is a generalization of 

a result already pointed out in the case of a> = 0, viz. that in oscillations about 

absolute equilibrium the mean values of the kinetic and potential energies 

are equal. 

The present result can be expressed in another form. If for a moment we 

regard a as a function of A r , B r , where these coefficients have general values, 

determined by the equation 

a*P + aQ-R=0, (10) 

we have 

(2o-P+Q)A<r + (<r 2 AP + o-AQ-A£)==0 (11) 

Hence when A r , B r have the special values appropriate to a free mode of 

oscillation, we have 

A<r = 0, (12) 

by (8). In other words, the values of a determined by (10) are stationary. 

It follows that if the values of P, Q, R in (10) are calculated on the basis 
of an assumed type of vibration which differs slightly from the truth, the 
error in the consequent values of a will be of the second order. 

These stationary values will include, as generally most important, the 
maxima and minima (in absolute value) of a. 

Applications of the above principle to particular cases will be found in 
Arts. 212 a, 216. 

The general form of the functions P, Q, R in (7) may be noticed, although 
it is not essential to the argument. We have at once, on reference to 
Art. 204 (3) (4), 

P = iS r a r (A r 2 + B r 2 ) } R = lS r c r {A* + B r 2 ), (13) 

where S r denotes a summation of terms of the types indicated, with r = 1 , 2, . . . n. 
Again, from (2), 

= \ {qxSrPlrqr + 22#r£2r?r + • • • + ?„# r /3 nr a r }, (14) 

where 

ft,-2.S«l^ (15) 

o{q ai q r ) 

Substituting from (6), and taking uhe mean value, we have 

Q = \8 T a,fi n A,B r , (16) 

where, in the double summation, each permutation of suffixes is to be taken 
once. 

As a verification we may note that if with these values of P, Q, R we form 
the equation (8) the coefficients of A^l r , AP r will be found to be identical with 
the coefficients of cos at and sin at, respectively, when we substitute from (6) 
in the typical equation of motion, Art. 204 (5). 



(01Y 



316 Tidal Waves [chap, viii 

206. The symbolical expressions for the forced oscillations due to a 
periodic disturbing force are easily written down. If we assume that 
Qi, Qz> • •• Qn all vary as e^ 1 , where a is prescribed, the equations (5) of Art. 
204 give, if we omit the time-factors, 

D(i<r)q r = a rl Q 1 + a r2 Q 2 + ...+a™Qn> (1) 

where the coefficients on the right-hand side are the minors of the rth row 
in the determinant D (ia). 

The most important point of contrast with the theory of the 'normal 
modes' in the case of no rotation is that the displacement of any one type 
is no longer affected solely by the disturbing force of that type. As a con- 
sequence, the motions of the individual particles are, as is easily seen from 
Art. 205 (14), now in general elliptic-harmonic. Again, there are in general 
differences of phase, variable with the frequency, between the displacements 
and the force. 

As in Art. 168, the displacement becomes very great when D (ia) is very 
small, i.e. whenever the 'speed* <r of the disturbing force approximates to 
that of one of the natural modes of free oscillation. 

When the period of the disturbing forces is infinitely long, the displace- 
ments tend to the 'equilibrium- values' 

qi = Qi/ci, g 2 = Q 2 /c 2 , ... q n = Qnlc n , (2) 

as is seen directly from the equations (5) of Art. 204. This conclusion must 
be modified, however, when one or more of the coefficients of stability c 1} 
c 2 , ... c n is zero. If, for example, Ci = 0, the first row and column of the deter- 
minant D (X) are both divisible by \, so that the determinantal equation 
has a pair of zero roots. In other words we have a possible free motion of 
infinitely long period. The coefficients of Q 2 , Q3, ••• Qn on the right-hand 
side of (1) then become indeterminate for a — 0, and the evaluated results 
do not as a rule coincide with (2). This point is of importance, because in 
some hydrodynamical applications, as we shall see, steady circulatory motions 
of the fluid, with a constant deformation of the free surface, are possible when 
no extraneous forces act; and as a consequence forced tidal oscillations of 
long period do not necessarily approximate to the values given by the equi- 
librium theory of the tides. Cf. Arts. 214, 217. 

In order to elucidate the foregoing statements we may consider more in detail the case 
of two degrees of freedom. The equations of motion are then of the forms 

«i?i + cig'i+/3£ 2 = # 1 , a 2 J2 + c 2 g'2-0yi = #2 ( 3 ) 

The equation determining the periods of the free oscillations is 

ai«2X 4 + (aiC 2 + a 2 c 1 +/3 2 )X 2 +CiC 2 =0 (4) 

For ' ordinary ' stability it is sufficient that the roots of this quadratic in X 2 should be real 
and negative. Since a x , a 2 are essentially positive, it is easily seen that this condition is 
in any case fulfilled if c x , c 2 are both positive, and that it will also be satisfied even when 



206-207] Forced Oscillations 317 

0|, c 2 are both negative, provided £ 2 be sufficiently great. It will be shewn later, however, 
that in the latter case the equilibrium is rendered unstable by the introduction of dissipa- 
tive forces. See Art. 322. 

To find the forced oscillations when Q u Q 2 vary as e* "', we have, omitting the time- 
factor, 

(c 1 -a 2 a 1 )q 1 + iorPq 2 =Qi, - i<rpqi+(c 2 - <r 2 a 2 ) q 2 =Q 2 , (5) 

, onpfl n _ (02-0*02) Qi-i<rpQ 2 _ ^^i + (ci-q- 2 ai)^ 2 , fi s 

wnence q x - ^ _ ^ ^ _ ^ _ ^ , q 2 ^ _ ^ ^ _ ^ _^ 2 W 

Let us now suppose that c 2 = 0, or, in other words, that the displacement q 2 does not 
affect the value of V— T . We will also suppose that Q 2 =0, i.e. that the extraneous forces 
do no work during a displacement of the type q 2 . The above formulae then give 

q ^a 2 (c 1 -a 2 a 1 )+^ ^ = a 2 (c 1 -a*a 1 )+^ Ql (7) 

In the case of a disturbance of long period we have o-=0, approximately, and therefore 

a-s^s*' ^^+w Qi (8) 

The displacement q x is therefore less than its equilibrium -value, in the ratio 1 : 1 +/3 2 /«2<?i ; 
and it is accompanied by a motion of the type q 2 although there is no extraneous force of 
the latter type (cf. Art. 217). We pass, of course, to the case of absolute equilibrium, 
considered in Art. 168, by putting /3 = 0*. 

It should be added that the determination of the 'principal co-ordinates' 
of Art. 204 depends on the original forms of f& and V — T , and is therefore 
affected by the value of co 2 , which enters as a factor of T . The system of 
equations there given is accordingly not altogether suitable for a discussion 
of the question how the character and the frequencies of the respective 
principal modes of free vibration vary with co. One remarkable point which 
is thus overlooked is that types of circulatory motion, which are of infinitely 
long period in the case of no rotation, may be converted by the slightest 
degree of rotation into oscillatory modes of periods comparable with that of 
the rotation. Cf. Arts. 212, 223. 

To illustrate the matter in its simplest form, we may take the case of two degrees of 
freedom. If c 2 vanishes for o>=0, and so contains o> 2 as a factor in the general case, the 
two roots of equation (4) are 

X 2 = - c x \a x , X 2 = - c 2 /a 2 , 

approximately, when o> 2 is small. The latter root makes X <x co, ultimately. 

207. Proceeding to the hydrodynamical examples, we begin with the case 
of a plane horizontal sheet of water having in the undisturbed state a motion 
of uniform rotation about a vertical axisf. The results will apply without 
serious qualification to the case of a polar or other basin, of not too great 
dimensions, on a rotating globe. 

* The preceding theory appeared in the 2nd ed. (1895) of this work. The effect of friction is 
considered in Art. 322. 

t Sir W. Thomson, "On Gravitational Oscillations of Rotating Water," Proc. R. S. Edin. 
x. 92 (1879) [Papers, iv. 141]. 



318 Tidal Waves [chap, viii 

Let the axis of rotation be taken as axis of z. The axes of x and y being 
now supposed to rotate in their own plane with the prescribed angular 
velocity co, let us denote by u, v, w the velocities at time t, relative to these axes, 
of the particle which then occupies the position (x, y, z). The actual velocities 
of the same particle, parallel to the instantaneous positions of the axes, will 
be u — ayy, v + cox, w, and the accelerations in the same directions will be 

Du _ o Bv a „ Dw 

^ -&*-«*, JJJ + 2.U-.V, -jj t - 

In the present application, the relative motion is assumed to be infinitely 
small, so that we may replace D/Dt by d/dt. 

Now let z be the ordinate of the free surface when there is relative 
equilibrium under gravity alone, so that 

2 

zo = i — (# 2 + y z ) 4- const., (1) 

as in Art. 26. For simplicity we will suppose that the slope of this surface 
is everywhere very small ; in other words, if r be the greatest distance of any 
part of the sheet from the axis of rotation, co 2 r/g is assumed to be small. 

If z +£ denote the ordinate of the free surface when disturbed, then on 
the usual assumption that the vertical acceleration of the water is small 
compared with g, the pressure at any point (x, y, z) will be given by 

p-Po = gp(*o+S-*)> (2) 

, ldp 2 3f I dp 2 dt 

whence ^=-co*x-g^-, --£■ = - co 2 y - g ^ . 

pdx a ox pdy dy 

The equations of horizontal motion are therefore 

du a? an dv i . a? an 

where ft denotes the potential of the disturbing forces. 

If we write ?= — &lff* (4) 

i.e. f is the ' equilibrium ' value of the surface elevation, these become 

I-*—- '£«-?>. S +1 — -4 (t -- B (5) 

The equation of continuity has the same form as in Art. 193, viz. 

3C_ d(hu) aw 

dt dx dy ' { } 

whei-e h denotes the depth, from the free surface to the bottom, in the 
undisturbed condition. This depth will not, of course, be uniform unless the 
bottom follows the curvature of the free surface as given by (1). 



207-208] Plane Sheet of Water 319 

If we eliminate £-£from the equations (5), by cross-differentiation, we find 

1(1-1)^(1-1)=°' <» 

or, writing u = d^/dt, v = drjldt, and integrating with respect to t, 

s-5+-ffi+8— - ;-- (8) 

This is merely the expression of Helmholtz' theorem that the product of the vorticity 

2o> + ^- - y and the cross-section ( 1 + ~- + J- j §# dy, 

of a vortex-filament, is constant. 

In the case of a simple-harmonic disturbance, the time-factor being e wt 
the equations (5) and (6) become 

iau-2(DV = -g^(£-£), i*v+ 2ayu= -#;-(£- ?), (9) 

. d(hu) d(hv) m 

and ^ = -"^ ^T (10) 

From (9) we find 

(11) 

and if we substitute from these in (10), we obtain an equation in f only. 
In the case of uniform depth the result takes the form 

v 1 2 r+ <T ^^=v 1 ^ (12) 

where V 2 2 = d 2 /da? + d 2 /dy 2 , as before. 

When £=0, the equations (5) and (6) can be satisfied by constant values of u, v, £ 
provided certain conditions are fulfilled. We must have 

u--ff, v=£f ,.(13) 

2o> dy ' 2a> ex 

aDd therefore \ [ h > ^ =0 (14) 

d {x, y) 

The latter condition shews that the contour-lines of the free surface must be everywhere 
parallel to the contour-lines of the bottom, but that the value of £ is otherwise arbitrary. 
The flow of the fluid is everywhere parallel to the contour-lines, and it is therefore further 
necessary for the possibility of such steady motions that the depth should be uniform 
along the boundary (supposed to be a vertical wall). When the depth is everywhere the 
same, the condition (14) is satisfied identically, and the only limitation on the value of f 
is that it should be constant along the boundary. 



A simple application of the preceding equations is to the case of 
free waves in an infinitely long uniform straight canal*. 

If we assume f =sae iic(ct-x)+my j v = 0, (1) 

the axis of x being parallel to the length of the canal, the equations (5) of the 
preceding Art., with the terms in f omitted, give 

cu = g£, 2(au = -gm£, (2) 

* Sir W. Thomson, I.e. ante p. 317. 



320 Tidal Waves [chap, vm 

whilst, from the equation of continuity (Art. 207 (6)), 

c£=hu (3) 

We thence derive c 2 = gh, m = — 2co/c (4) 

The former of these results shews that the wave-velocity is unaffected by the 
rotation. 

When expressed in real form, the value of £ is 

f =ae- 2 »yl c cos {k(ct-x) + e} (5) 

The exponential factor indicates that the wave-height increases as we pass from 
one side of the canal to the other, being least on the side which is forward 
in respect of the rotation. If we take account of the directions of motion of 
a water- particle, at a crest and at a trough, respectively, this result is seen to 
be in accordance with the tendency pointed out in Art. 202*. 

It will be observed that there is, in the above solution, no limitation to 
the breadth of the canal, provided it be uniform. 

The problem of determining the free oscillations in a rotating canal of 
finite length, or even the simpler one of reflection of a wave at a transverse 
barrier, does not however admit of a simple solution by superposition, as was 
the case in the investigations of Arts. 176, 178. For a wave travelling in the 
negative direction, we should find 

f= ft ' 6 2# cos {k (ct + «) + €'}, (6) 

but this cannot be combined with (5) so as to make u — at a barrier for all 
values of yf. 

209. We take next the case of a circular sheet of water rotating about 
its centre}. 

If we introduce polar co-ordinates r, 6, and employ the symbols £, r\ to 
denote displacements along and perpendicular to the radius vector, then since 
f = iai;, r) = iar) i the equations (9) of Art. 207 are equivalent to 

a 2 ?+2 4 W^ = fi r|,(r-r) ) <T 2 ,-2»Wf -g±tf-Q, (1) 

* For applications to tidal phenomena see Sir W. Thomson, Nature, xix. 154, 571 (1879), and 
G. I. Taylor, "Tidal Friction in the Irish Sea," Phil. Trans. A, ccxx. 1 (1918). 

t Poincare\ Legons de Mec. Cel. iii. 124. The problem here indicated has been solved by 
G. I. Taylor, Proc. Lond. Math. Soc. (2) xx. 148 (1920). He finds that, provided the wave-length 
(27r/ft) be sufficiently large compared with the breadth (6), there is regular reflection (with a change 
of phase), in the sense that at a distance from the barrier we have practically superposition of 
(5) and (6) above, with a' = a, the necessary condition being 

fc 2 6 2 <7r 2 + 4w 2 & 2 /c 2 . 
The theory of the free oscillations in a rotating rectangular basin is also discussed in the paper 
cited. The case where the angular velocity of rotation is relatively small had been previously 
treated by Eayleigh, Phil. Mag. (6), v. 297 (1903) [Papers, v. 93], and Proc. Roy. Soc. A, lxxxii. 
448 (1909) [Papers, v, 497J. 

J The investigation which follows is a development of some indications given by Kelvin in 
the paper cited on p. 317. 



208-210] Rotating Circular Basin 321 

whilsb the equation of continuity (10) becomes 

d(h&) d(h v ) 

6 rdr rdd V } 

Hence 

(3) 

and substituting in (2) we get the differential equation in f. 

In the case of uniform depth we find 

(v^+^jr-v^g (4) 

n2 a 2 i d i a 2 /KX 

where ^=- 2 + - 8 - + - 2 -- 2 , (5) 



a 2 -4 



and /c 2 = j — (6) 

gh 

This might have been written down at once from Art. 207 (12). 

The condition to be satisfied at the boundary (r = a, say) is f = 0, or 

('!-*&«-»-<> < 7 > 

210. In the case of the free oscillations we have f=0. The way in which 
the imaginary i enters into the above equations, taken in conjunction with 
Fourier's Theorem, suggests that occurs in the form of a factor e is9 , where s 
is integral. On this supposition, the differential equation (4) becomes 

a 2 ? 13? / 2 s 2 \ 

d? + rdr + [ K -r^ = °' (8) 

and the boundary-condition (7) gives 

'g+^=o> < 9 > 

for r = a. 

The equation (8) is of Bessel's form, and the solution which is finite for 
r = may therefore be written 

^AJ.Ur)^^^; .. (10) 

but it is to be noticed that k 2 is not, in the present problem, necessarily 
positive. When k 2 is negative, we may replace J 8 (/cr) by I s (/cir), where ki is 
the positive square root of (4&> 2 — cfi)lgK and 

Z s ( Z 2 2 4 



Ia ^ 2 s . 5 ! l 1 + 2(2 5 + 2) + 2.4(2.5 + 2)(2s + 4) + ---j* "'^ U ) 
In the case of symmetry about the axis (5 = 0), we have, in real form, 

£= A Jo(tcr). coa (<rt + €), (12) 

* The functions I a (z) were tabulated by Prof. A. Lodge, Brit. Ass. Rep. 1889. The tables are 
reprinted by Dale, and by Jahnke and Emde. Extensive tables of the functions e~-*I (z), e~ z I x (z) 
are given in Watson's treatise. 



322 Tidal Waves [chap, viii 

where k is determined by 

J ' (*a) = (13) 

The corresponding values of a are then given by (6). The free surface has, 
in the various modes, the same forms as in Art. 191, but the frequencies are 
now greater. If we write 

c 2 = gh, /3 = 4,co 2 a 2 /c 2 , (14) 

we have o 2 a 2 \c 2 = K 2 a 2 + /3 (15) 

It is easily seen, moreover, on reference to (3), that the relative motions of 
the fluid particles are no longer purely radial ; the particles describe, in fact, 
ellipses whose major axes are in the direction of the radius vector. 

For s > we have 

f =A J 8 (tcr). cos (at + sO + e), (16) 

where the admissible values of k, and thence of a, are determined by (9), 
which gives 

KaJ s '(tca) + — J s (/ea) = (17) 

G 

The formula (16) represents a wave rotating relatively to the water with 
an angular velocity cr/s, the rotation of the wave being in the same direction 
with that of the water, or the opposite, according as a/a> is negative or 
positive. 

If ica is any real or pure imaginary root of (17), the corresponding value 
of a is given by (15). 

Some indications as to the values of <r may be gathered from a graphical construction. 
If we write K 2 a 2 = x, we have, from (6), 

£=±H)* ™ 

If we further put *ffi g \ = <t> (k 2 « 2 ), 

KOJ s {kO) r X 

the equation (17) may be written 

0(^)±fl+f N ) =0 (19) 



The curve #=-<£(#) (20) 

can be readily traced by means of the tables of the functions J 8 (z), I 8 (z) ; and its inter- 
sections with the parabola 

y 2 = l+^/3 (21) 

will give, by their ordinates, the values of ct-/2co. The constant /3, on which the positions 
of the roots depend, is equal to the square of the ratio 2coa/(gh)^ which the period of a 
wave travelling round a circular canal of depth h and perimeter 2ira bears to the half- 
period (jt/o>) of the rotation of the water. 

The diagrams on the next page indicate the relative magnitudes of the lower roots, in 
the cases s=l and s = 2, when /3 has the values 2, 6, 40, respectively*. 

* For clearness the scale of y has been taken to be 10 times that of x. 



210] 



Free Oscillations 



323 



With the help of these figures we can trace, in a general way, the changes in the 
character of the free modes as /3 increases from zero. The results may be interpreted as 
due either to a continuous increase of co, or to a continuous diminution of h. We will use 




[•-«] 



324 



Tidal Waves 



[chap. VIII 



the terms 'positive' and 'negative' to distinguish waves which travel, relatively to the 
water, in the same direction as the rotation and the opposite. 

When /3 is infinitely small, the values of x are given by J 8 ' (#z)=0; these correspond 
to the vertical asymptotes of the curve (20). The values of cr then occur in pairs of equal 
and oppositely- signed quantities, indicating that there is now no difference between the 
velocity of positive and negative waves. The case is, in fact, that of Art. 191 (12). 

As /3 increases, the two values of a- forming a pair become unequal in magnitude, and 
the corresponding values of x separate, that being the greater for which o-/2a> is positive. 
When /3=s (s + 1) the curve (20) and the parabola (21) touch at the point (0, -1), the 
corresponding value of o- being — 2a>. As /3 increases beyond this critical value, one value 
of x becomes negative, and the corresponding (negative) value of o-/2a> becomes smaller 
and smaller. 

Hence, as /3 increases from zero, the relative angular velocity becomes greater for a 
negative than for a positive wave of (approximately) the same type ; moreover the value 
of a for a negative wave is always greater than 2w. As the rotation increases, the two 
kinds of wave become more and more distinct in character as well as in ' speed.' With a 
sufficiently great value of /3 we may have one, but never more than one, positive wave for 
which cr is numerically less than 2o>. Finally, when /3 is very great, the value of o- corre- 
sponding to this wave becomes very small compared with 2o>, whilst the remaining values 
tend all to become more and more nearly equal to + 2o. 



If we use a zero suffix to distinguish the case of w = 0, we find 

o- 2 _ K 2 + 4co 2 /gh _ x + p 



.(22) 



where x refers to the proper asymptote of the curve (20). This gives the 'speed' of any 
free mode in terms of that of the corresponding mode when there is no rotation. 

The preceding statements are illustrated by the following table, which gives for the 
case of s = 1 approximate values of <a within the range of the upper diagram on p. 323, 
together with the corresponding values of o-/2o> and aa/c. 



(3 = 


,8=2 


,3=6 


j8 = 40 


j8 = co 


Ka = aa/c 


Ka 


<r/2w 


a-a/c 


Ka 


<r/2o> 


<xa/c 


Ka 


o-/2w 


<rajc 


Ka 


(r/2w 


<ra\e 


±1-84 
±533 


J2-19 

I o 

(5-38 
(5-28 


+ 1-84 
-1-00 
+ 3-93 
-3-86 


+ 2-61 
-1-41 
+ 5-56 
-5-47 


J2-29 
\2-10i 
p-41 
(5-25 


+ 1-37 
-0-51 

+ 2-42 
-2-37 


+ 3-35 

-1-26 
+ 5-94 
-5-79 


J2-38 
[6'2Si 
J5-47 
\5-18 


+ 1-07 
-0-17 
+ 1-32 
-1-29 


+ 6-76 
-1-09 
+ 8-36 
-8-17 


T2-40 

J5-52 
(5-14 


+ 1-00 

+ 1-00 
-1-00 


-l'OO 

+0* 

-0* 



211. As a sufficient example of forced oscillations we may assume 



i-O 



,i(<rt + s9+e) 



.(23) 



where the value of a is now prescribed. 

This makes V x 2 f = 0, and the equation (4) then gives 
f=4/ s (*r)^+^ +e ), 



.(24) 



2io-2ii] Forced Oscillations 325 

where A is to be determined by the boundary-condition (7), viz. 

2^ 



s 1 + 



i") 



^= i — ^ .a (25) 

T . , . , 2«ft) r / \ 

/eaJ g («a) H J s (/ea) 

a* 



This becomes very great when the frequency of the disturbance is nearly 
coincident with that of a free mode of corresponding type*. 

From the point of view of tidal theory the most interesting cases are those of s = 1 
with o- = g), and s = 2 with <r=2o), respectively. These would represent the diurnal and 
semidiurnal tides due to a distant disturbing body whose proper motion may be neglected 
in comparison with the rotation <o. 

In the case of s = 1 we have a uniform horizontal disturbing force. Putting, in addition, 
<r = (o, we find without difficulty that the amplitude of the tide-elevation at the edge (r = a) 
of the basin has to its 'equilibrium-value' the ratio 

/i(i)+i4(i)' K } 

where 2=^(30). With the help of Lodge's tables we find that this ratio has the values 

1-000, -638, -396, 

for /3= 0, 12, 48, respectively. 

When <r = 2o), we have k=0, and thence, by (23), (24), (25), 

C = l (27) 

i.e. the tidal elevation has exactly the equilibrium-value. 

This remarkable result can be obtained in a more general manner ; it holds whenever 
the disturbing force is of the type 

J =x ( r ) c i(2«t + tf + e) ( 28 ) 

provided the depth h be a function of r only. If we revert to the equations (1), we notice 
that when o- = 2a> they are satisfied by £= £, r) = i%. To determine £ as a function of r, we 
substitute in the equation of continuity (2), which gives 

f-^=-xW (29) 

The arbitrary constant which appears on integration of this equation is to be determined 
by the boundary-condition. 

In the present case we have x(r) = 8 / a *- Integrating, and making £ = for r = a, 
we find 

Or 8 ' 1 
A£=~T- ( a 2-r 2 )e*( 2 ^ + 8fl+ <) (30) 

The relation r) = ig shews that the amplitudes of £ and r\ are equal, while their phases 
differ by 90° ; the relative orbits of the fluid particles are in fact circles of radii 

Cr*- 1 

r= u^^-^ < 31 > 

described each about its centre with angular velocity 2a> in the negative direction. We 
may easily deduce that the path of any particle in space is an ellipse of semi-axes r±r 
described about the origin with harmonic motion in the positive direction, the period 
being 27r/©. This accounts for the peculiar features of the case. For if £ have always the 

* The case of a nearly circular sheet is treated by Proudman, "On some Cases of Tidal Motion 
on Eotating Sheets of Water," Proc. Lond. Math. Soc. (2) xii. 453 (1913). 



326 Tidal Waves [chap, viii 

equilibrium-value, the horizontal forces due to the elevation exactly balance the disturbing 
force, and there remain only the forces due to the undisturbed form of the free surface 
(Art. 207 (1)). These give an acceleration gdzjdr, or o>V, to the centre, where r is the 
radius vector of the particle in its actual position. Hence all the conditions of the problem 
are satisfied by elliptic-harmonic motion of the individual particles provided the positions, 
the dimensions, and the 'epochs' of the orbits can be adjusted so as to satisfy the con- 
dition of continuity, with the assumed value of £. The investigation just given resolves 
this point. 

When the sheet of water is bounded also by radial walls the problem is more difficult. 
The tidal oscillations (free and forced) in a semicircular basin of uniform depth are discussed 
by Proudman*, with an application to the tides of the Black Sea, the disturbing forces 
being of the idealized diurnal and semi-diurnal types. 

The free and forced oscillations in a rotating elliptic basin of uniform depth are discussed 
by Goldstein f. 

212}. We may also notice the case of a circular basin of variable depth, 
the law of depth being the same as in Art. 193, viz. 



h 



= Ao ( 1 -^ « 



Assuming that £, rj, £ all vary as e i ( <rt + «* +€ ), and that A is a function of r only, we 
find, from Art. 209 (2), (3), 

--v>c + .g(j + Stt-a + *(S + i4-aK-ft-o m 



(i-S)0+S-SO-5(^ + ^) + -- f - 



( 

Introducing the value of h from (1), we have, for the free oscillations, 

o- 2 -4o) 2 
9h 
This is identical with Art. 193 (6), except that we now have 

o- 2 — 4co 2 4cos 
gk va 2 

in place of (r 2 jgk . The solution can therefore be written down from the results of that 
Art., viz. if we put 

(<r 2 -4co 2 )a 2 4cos , . „ 

— PT — = n{n-Z)-s\ (4) 

we have f=4,QVL^7 ) ^e i ( <rt + 8( ' + e ), (5) 

where a=%n+§s, /3=l+iU-|rc, y = s + l; 

and the condition of convergence at the boundary r = a requires that 

n = s + 2j, (6) 

where,;' is some positive integer. The values of a are then given by (4). 

The forms of the free surface are therefore the same as in the case of no rotation, but 
the motion of the water particles is different. The relative orbits are in fact now ellipses 
having their principal axes along and perpendicular to the radius vector; this follows 
easily from Art. 209 (3). 

* M. N. R. A. S., Geophys. Suppt. ii. 32 (1928). 

t Ibid. ii. 213 (1929). 

X See the footnote to Art. 193. 



2ii-2i2] Basin of Variable Depth 327 

In the symmetrical modes (* = 0), the equation (4) gives 

0-2 = ^2 + 4^ ( 7 ) 

where <r denotes the ' speed ' of the corresponding mode in the case of no rotation, as 
found in Art. 193. 

For any value of s other than zero, the most important modes are those for which 
n = s + 2. The equation (4) is then divisible by o- + 2o>, but this is an extraneous factor ; 
discarding it, we have the quadratic 

o- 2 -2o>o-=2s^°, (8) 

a 1 

whence o-=<*± L*+2s 9 -^y . (9) 

This gives two waves rotating round the origin, the relative wave-velocity being greater 
for the negative than for the positive wave, as in the case of uniform depth (Art. 210). 
With the help of (8) the formulae reduce to 

<-*©'• e-*^©'"'. " =ii i A -&" 1 ' (10) 

the factor e l i <rt + s6 + e ) being understood in each case. Since 17— *£, the relative orbits are 
all circles. 

The case 5=1, n = S, is noteworthy; the free surface is then always plane, and the 
circular orbits have all the same radius. In the following table, which relates to this case, 
/3 stands for 4o 2 a 2 /c 2 , where c = *J(gh ). 



p=o 


,3=2 


£=6 


/3 = 40 


aajco 


tr/2w 


<rajc 


<r/2w 


<ra/c 


<r/2w 


<ra/c 


±1-414 


+ 1-618 
-0-618 


+ 2-288 
-0-874 


+ 1-264 
-0-264 


+ 3-096 
- 0-646 


+ 1 -048 
-0-048 


+ 6-626 
-0-302 



When ?i>5 + 2, we have nodal circles. The equation (4) is then a cubic in o-/2to; it is 
easily seen that its roots are all real, lying between - co and -1,-1 and 0, and +1 and 
+ oo , respectively. The following table is calculated for the case of s = 1, n = 5. 



,8=0 


i 
P = 2 13 = 6 

I 


/3 = 40 


<xa/c 


<r/2w 


cra/co 


<r/2w 


aajc 


cr/2w 


<ralc 


±3-742 


+ 2-889 
-0-125 

-2-764 


+ 4-085 
-0-176 
-3-909 


+ 1-874 
-0-100 
-1-774 


+ 4-590 
-0-245 
-4-344 


+ 1-183 
- 0-040 
-1-143 


+ 7-483 
-0-253 
-7-230 



The first and the last root of each triad give positive and negative waves of a somewhat 
similar character to those already obtained in the case of uniform depth. The smaller 
negative root gives a comparative slow oscillation which, when the angular velocity o> is 
infinitely small, becomes a steady rotational motion, without elevation or depression of the 
surface. The possibility of oscillations of this type was pointed out in Art. 206, ad fin. In 



328 Tidal Waves [chap, viii 

the present case the transition is easily traced. It follows from (4) that the relevant limiting 
value of o-/2co, when co is infinitesimal, is - } . We then find, from Art. 209 (2), (3), 

^cfl-^jeHO + 't), ^wU-bfSeW*^ (11) 

with £= -*^r (l - f -f) e i( ' + <r<) , (12) 

ultimately, where <r=— fco. 

The most important type of forced oscillations is such that 

l=C (-Ye^+80 + e) (13) 

We readily verify, on substitution in (3), that 



*" 2sgh -(v*-2a>o-)a 2 ^ ^ 

We notice that when a = 2a> the tide-height has exactly the equilibrium-value, in agree- 
ment with Art. 211. 

If <r 1? o- 2 denote the two roots of (8), the last formula may be written 

^(l-er/erxXl-T/^) (15) 

The tidal oscillations in a semicircular basin with the above law of depth have been 
examined by Goldsbrough *. The difficulty of the problem consists in satisfying the 
conditions at the straight portion of the boundary. 

212 a. Place may be found here for one or two illustrations of the approxi- 
mate procedure outlined in Art. 205 a. 

1°. To take first a known problem, that of the circular basin of uniform depth (Art. 
210). Assuming as the polar co-ordinates of a displaced particle, relative to an initial line 
revolving with the angular velocity o>, 

r'=r+& 6' = 0+T)/r, (1) 

the equation of continuity is 

h dr r rd0 ' 
as in Art. 209 (2). 

With our previous notation 



•(2) 



(3) 



/a f2n fa f2ir "\ 

J H'+Wrdddr, r-T =igpJJ o ?rd6dr, 

M> = phj a J^{^-4)rdedr. 
We take as our assumed type, for the gravest mode, 

$ = A (l - ^ cos (*t + 6), V = (-A + B r ^ sin(o-< + 0), (4) 



which make 



jr=(3A-B)^cos{<rt + 0) (5) 



The constants in (4) have been adjusted so that £ shall be finite for r — 0. 

* Proc. Roy. Soc. cxxii. 228 (1929). 



212-212 a] Approximate Method 329 

Hence with the definitions of Art. 205 a, taking the mean values of the functions in 
(3), and performing the integrations, 

P=^7rpha 2 (4:A 2 -SAB+B 2 ) } Q= -lirpcohcfi^A 2 - AB), R=%7rgph 2 (3A-B) 2 . ...(6) 

If we write for shortness 

c=s/feA)i <ra/c = x, 4a> 2 a 2 /c 2 = 0, (7) 

the equation 

(r 2 P+aQ-R = (8) 

becomes 

(4x 2 -3j(3x-\ 7 -)A 2 -(Sx 2 - s /(3x-9)AB + (x 2 -%)B 2 = (9) 

The stationary values of uc are then given by 

x 2 (7x 2 - 6Vj8a?-/3- 24) = (10) 

The zero roots may be disregarded as corresponding to a merely circulatory motion, without 
change of surface-level. To compare with the numerical results of Art. 210 (p. 324) we 
put /3 = 2, 6, 40 in succession. The finite roots of (10) are 



-1-43) -1-271 

+ 2-65J' +3-27] ' 



-1-35) 

+ 6-77 



in the respective cases. It is only in the third case that there is any serious deviation 
from the correct value. It will be seen that the approximate method is fairly successful 
over a considerable range of the parameter /3. 

2°. In the case of a rectangular basin of uniform depth, we take axes Ox, Oy coincident 
with two of the sides, whose lengths are (say) a, b, respectively. Denoting by £, 77 the com- 
ponent displacements of a particle, we have 



Z=WJ a J\p+v*)<tedy, V-T»=\9P j'J^dxdy^ 

Let us assume as an approximate type 



/ (&l-rih)dxdy. 



.(11) 



£ == A sin — cos a-t, rj = B sin -~ sin at (12) 

a 

This is suggested by the case of o> = 0, where either A or B is zero, and cannot be expected 
to give a good result for more than a limited range of a>. From (12) we derive 

C d£ dn (A ttx B try . \ ,„. 

y = - ~ -7T = — w — cos — cos o-*+ t-cos-/- sin at) (13) 

h dx dy \a a b J v ' 

Hence 

P=\phab(A*+&\ qJ^^AB, R=\* 2 gph 2 (±A 2 + \B^ (14) 

The equation (8) now takes the form 

(<r>-^)A* + ?^AB + ^-^)B> = 0, (15) 

where c 2 =gh as before. The stationary values of a are therefore given by 

(^-<^W)= 2 -^-\ (i6) 

where <r u a 2 are the values of a corresponding to oscillations parallel to x and y, respec- 
tively, when there is no rotation. 



330 Tidal Waves [chap, viii 

If a) is small and a, b decidedly unequal, then in the type where <r = o- x , nearly, we have 



128c 



(T- 0"i= -47- o 2\) \ 1 ' ) 

7r 4 (cr^-(r2 2 ) 



approximately. The corresponding ratio BjA is then given by 



7T 

and is accordingly small, as was to be expected. 

For a square tank (a = b), on the other hand, (16) makes 



ISapA+W-flB-O, (18) 



. 2 _ . 1 2 =± — , 9) 

'-n-±% (*» 



approximately. Then B/A = + 1 . 

Tides on a Rotating Globe. 

213. We proceed to give some account of Laplace's problem of the tidal 
oscillations of an ocean of (comparatively) small depth covering a rotating 
globe *. In order to bring out more clearly the nature of the approximations 
which are made on various grounds, we adopt a method of establishing the 
fundamental equations somewhat different from that usually followed. 

When in relative equilibrium, the free surface is of course a level-surface 
with respect to gravity and centrifugal force ; we shall assume it to be a 
surface of revolution about the polar axis, but the ellipticity will not in the 
first instance be taken to be small. 

We adopt this equilibrium-form of the free surface as a surface of reference, 
and denote by and <j> the co-latitude (i.e. the angle which the normal makes 
with the polar axis) and the longitude, respectively, of any point upon it. 
We shall further denote by z the altitude, measured outwards along a normal, 
of any point above this surface. 

The relative position of any particle of the fluid being specified by the 
three orthogonal co-ordinates 0, <f>, z, the kinetic energy of unit mass is 
given by 

27 7 =(i2 + *) 2 <9 2 +GT 2 (a> + <£) 2 -fi 2 (1) 

where R is the radius of curvature of the meridian-section of the surface of 
reference, and ta is the distance of the particle from the polar axis. It is to 
be noticed that R is a function of only, while -or is a function of both 
and z ; and it easily follows from geometrical considerations that 

= cos 0, «- = sin (2) 



(R + z)d0 ' dz 

* "Recherches sur quelques points du systeme du monde," Mem. de V Acad. roy. des Sciences, 
1775 [1778] and 1776 [1779]; Oeuvres Completes, ix. 88, 187. The investigation is reproduced, 
with various modifications, in the Mecanique Ctleste, Livre 4 me , c. i. (1799). 



212 a-213] 



Laplace's Theory of the Tides 



331 



The component accelerations are obtained at once from (1) by Lagrange's 
formula. Omitting terms of the second order, on account of the restriction to 
infinitely small motions, we have 



R+z\dtcQ ddj v 



R 



(co 2 + 2 G >4>)a>~ 



i{jt d ^- d i)=^ +2a} { 









fc 



■)■ 



(3) 



^8i-S = '- (< ° 2+2 ^ )CT ^ 



=- — 2&W COS 

Oh 



dv 



+ 2am cos 6 + 2&>w sin 






(5) 



Hence, if we write a, v, w for the component relative velocities of a particle, 
viz. 

U = (R+ z)0, V = 'BT<j), w=z } (4) 

and make use of (2), the hydrodynamical equations may be put in the forms 

1 d_ 

dt t»Td<t)\p 

— — 2&>v sin = — 

dt oz \p 

where M* is the gravitation-potential due to the earth's attraction, whilst H 
denotes the potential of the disturbing forces. 

So far the only approximation has consisted in the omission of terms of 
the second order in u, v, w. In the present application, the depth of the sea 
being small compared with the dimensions of the globe, we may replace 
R + z by R. We will further assume that the vertical velocity w is small 
compared with the horizontal components u, v and that dw/dt may be 
neglected in comparison with cov. As in the theory of 'long' waves, such 
assumptions are justified a posteriori if the results obtained are found to be 
consistent with them (cf. Art. 172)*. 

Let us integrate the third of equations (5) between the limits z and f, 
where f denotes the elevation of the disturbed surface above the surface of 
reference. At the surface of reference (z = 0) we have 

"9 — ^co 2 ™ 2 = const., 

by hypothesis, and therefore at the free surface (z = f ) 

*& — \cd 2 ^ 2 = const. + #£ 

8 



approximately, provided g = 



dz 



(¥- ico 2 ^ 2 ) 



z=0 



•(6) 



* Thus in the simplified conditions of Arts. 219, 220 wjwv is of the order m( = «%/#). 



332 Tidal Waves [chap, viii 

Here g denotes the value of apparent gravity at the surface of reference ; 
it is of course, in general, a function of 6, but its variation with z is 
neglected. 

The integration in question then gives 

^ + ^-ift> 2 OT 2 = const. +#?+2G>sin0 | vdz, (7) 

p Jz 

where the variation of the disturbing potential CI with z has been neglected 
in comparison with g. The last term is of the order of a>hv sin 6, where h is the 
depth of the fluid, and it may be shewn that in the subsequent applications 
this is of the order h/a as compared with g£ *. Hence, substituting in the first 
two of equations (5), we obtain, with the approximations indicated, 

where f =~n/# (9) 

These equations are independent of z, so that the horizontal motion may be 
assumed to be sensibly the same for all particles in the same vertical line. 

As in Art. 198, this last result greatly simplifies the equation of continuity. 
In the present case we find without difficulty 

dj_ 1 \d(hwu) d(hv)] 

dt~ v\ Rdd + d<t> j u; 

It is important to notice that the preceding equations involve no 
assumptions beyond those expressly laid down; in particular, there is no 
restriction as to the ellipticity of the meridian, which may be of any degree 
of oblateness. 

214. In order, however, to simplify the question as far as possible, with- 
out sacrificing any of its essential features, we now take advantage of the 
circumstance that in the actual case of the earth the ellipticity is a small 
quantity, being in fact comparable with the ratio (co 2 a/g) of centrifugal force 
to gravity at the equator, which ratio is known to be about ^J^. Subject to 
an error of this order of magnitude, we may put R = a, sr — a sin 6, g — const., 
where a is the earth's mean radius. We thus obtain 

^_2™cos0 = -^(f-f), |%2om cos = - 2 -JL-; (£-?), 

dt a oa s " dt asm6d<j> 

. „ (1) 

.,, 3? 1 (d(kuBind) , d(hv)) 
with £ = ; y ' + _A^l (2) 

dt a sin 6 { dO dcf> J v ; 

this last equation being identical with Art. 198 (l)f- 

* This, again, may be verified in the same cases. The upshot is that the vertical acceleration 
is neglected, as in the theory of 'long' waves. 

f Except for the notation these are the equations arrived at by Laplace, I.e. ante p. 330. 



213-215] Fundamental Equations 333 

Some conclusions of interest follow at once from the mere form of the 
equations (1). In the first place, if u, v denote the velocities along and 
perpendicular to any horizontal direction s, we easily find, by transformation 
of co-ordinates, 

g_2»vooefl — y|(?-f) (3) 

In the case of a narrow canal, the transverse velocity v is zero, and the 
equation (3) takes the same form as in the case of no rotation ; this has 
been assumed by anticipation in Art. 183. The only effect ot the rotation in 
such cases is to produce a slight slope of the wave-crests and furrows in the 
direction across the canal, as investigated in Art. 208. In the general case, 
resolving at right angles to the direction of the relative velocity (q, say), we 
see that a fluid particle has an apparent acceleration 2coq cos 6 towards the 
right of its path, in addition to that due to the forces. 

Again, by comparison of (1) with Art. 207 (5), we see that the oscillations 
of a sheet of water of relatively small dimensions, in co-latitude 6, will take 
place according to the same laws as those of a plane sheet rotating about 
a normal to its plane with angular velocity co cos 0. 

As in Art. 207, free steady motions are possible, subject to certain 
conditions. Putting f=0, we find that the equations (1) and (2) are 
satisfied by constant values of u, v, f, provided 

u ■ J. K v = $_ d I (4) 

2coa sin 6 cos d<f> ' 2coacos0d0' v 7 

dJ mw^ < 5 > 

The latter condition is satisfied by any assumption of the form 

?=/(Asec0), (6) 

and the equations (4) then give the values of n, v. It appears from (4) that 
the velocity in these steady motions is everywhere parallel to the contour- 
lines of the disturbed surface. 

If h is constant, or a function of the latitude only, the only condition 
imposed on f is that it should be independent of <£ ; in other words the 
elevation must be symmetrical about the polar axis. 

215. We shall suppose henceforward that the depth h is a function of 
only, and that the barriers to the sea, if any, coincide with parallels of 
latitude. 

We take first the cases where the disturbed form of the water-surface 
is one of revolution about polar axis. When the terms involving <f> 



334 Tidal Waves [chap, viii 

are omitted, the equations (1) and (2) of the preceding Art. take the 
forms 

™2ewcos0 = -2^(£ -■£)", ^ + 2cou cos = 0, (1) 

d£ d(hu sm0) 

With 37 = r— — -j- (2) 

dt a sin Odd v 

Assuming a time-factor e i<Tt , and solving for m, v, we find 

_ jgg ?/f_p> = _ 2&)(7 cos d ,j, p. /ox 

M ~o- a -4ft> 2 cos 2 <9a90 U gj ' ? o- 2 - 4 a> 2 cos 2 (9 a 90^ Q > '" {) 

.,, . o 9 (Aw sin 0) ,., 

with i(7?= ■ a-^ (4) 

a sin Odd 

The formulae for the component displacements (f, 77, say) can be written 
down from the relations u = j, = ^ r or w = icrf , = 1*770-. It appears that the 
fluid particles describe ellipses having their principal axes along the meridians 
and the parallels of latitude, respectively, the ratio of the axes being 
a/2co . sec 6. In the forced oscillations of the present type the ratio cr/2o) is 
very small ; so that the ellipses are very elongated, with the greatest length 
from E. to W., except in the neighbourhood of the equator. 

Eliminating u and v between (3) and (4), and writing, for shortness, 

<-*-r> £-■* T= m ' (5) 

-** ^(/^yiw=- 4 <- < 6 > 

In the case of uniform depth, this becomt 

k§&%)+ K ~-* : (7) 

where a = cos 0, and 3=—— == — — (8) 

h gh 

216. First, as regards the free oscillations. Putting £=0, we have 

*&$$+"-*> < 9 > 

and we notice that in the case of no rotation this is included in (1) of Art. 199, 
as may be seen by putting fif 2 = a 2 a 2 /gh, f= 00 . The general solution of (9) 
is necessarily of the form 

K=AF(j*) + Bf{ii). (10) 

where F(/jl) is an even, and f(p) an odd, function of fju, and the constants 
A, B are arbitrary. In the case of a zonal sea bounded by two parallels of 
latitude, the ratio A : B and the admissible values of / (and thence of the 
frequency cr/27r) are determined by the conditions that u = at each of these 
parallels. If the boundaries are symmetrically situated on opposite sides 



215-216] Case of Symmetry 335 

of the equator, the oscillations fall into two classes; viz. in one of these 
B = Q, and in the other A = 0. By supposing the boundaries to contract to 
points at the poles, we pass to the case of an unlimited ocean, and the 
admissible values of f are now determined by the condition that n must 
vanish for /x = ± 1. The argument is, in principle, exactly that of Art. 201, 
but the application of the last-mentioned condition is now more difficult, 
owing to the less familiar form in which the solution of the differential 
equation is obtained. 

In the case of symmetry with respect to the equator, we assume, following 
the method of Kelvin* and Darwin f, 

— j-r a d ^ = B lf , + B^+... + B 2j+1 ^^ + (11) 

This leads to 

f = .4 - ifBvf + i {Bi -f*B 3 ) p* + . . . + 1 (B+, -/» Vi) /** + •••. • • -(12) 

where A is arbitrary ; and makes 

^(^j^)-B 1 + S(B d -B 1 )^ + ...+(2j + l)(B ij+1 -B ij _ 1 )^+.... 

(13) 

Substituting in (9), and equating coefficients of the several powers of /j,, 
we find 

Bt-pA-O, (14) 

M'rfS*-* < 15 > 

and thenceforward 

B *"-{ 1 -y(ij+l)) B +*^3j(»j + l) B *-*- (16) 

These equations determine B\,Bz, ... B 2 j+i, ... in succession, in terms of 
A,a,nd the solution thus obtained would be appropriate, as already explained, 
to the case of a zonal sea bounded by two parallels in equal N. and S. latitudes. 
In the case of an ocean covering the globe, it would, as we shall prove, give 
infinite velocities at the poles, except for certain definite values of/. 

Let us write B 2 j + i/B 2 j-i — Nj+i ; (17) 

we shall shew, in the first place, that as j increases Nj must tend either to 
the limit or to the limit 1. The equation (16) may be written 

at i /3/ 2 | g 1 ns> 

iV ^ 1 - 1 2j(2j + l) + 2j(2j + l)N j (l *> 

* Sir W. Thomson, "Note on the 'Oscillations of the First Species ' in Laplace's Theory of 
the Tides," Phil. Mag. (4), 1. 279 (1875) [Papers, iv. 248]. 

f " On the Dynamical Theory of the Tides of Long Period," Proc. Roy. Soc. xli. 337 (1886) 
[Papers, i. 336]. 



336 Tidal Waves [chap, viii 

Hence, whenj is large, either 

^=W)' ; (19) 

approximately, or N j+1 is not small, in which case Nj+% will be nearly equal 
to 1, and the values of iV^+s, Nj+i, ... will tend more and more nearly to 1, 
the approximate formula being 

_ _ ^ =1 -§w^ (20) 

Hence, with increasing j, N t tends to one or other of the forms (19) and (20). 

In the former case (19), the series (11) will be convergent for ja — ± 1, and 
the solution will be valid over the whole globe. 

In the other event (20), the product N s iV 4 . . . N j+J , and therefore the 
coefficient i?2; + i, tends with increasing j to a finite limit other than zero. 
The series (11) will then, after some finite number of terms, become com- 
parable with 1 + (A 2 + fjL l + ..., or (1 -fi 2 )- 1 , so that we may write 

1 £-£ + t^'. (21) 



where L and M are functions of /x which remain finite when \x = ± 1. Hence, 
from (3), 

"■~ ^^f^-kv-^^-^M), -.(22) 
which makes u infinite at the poles. 

It follows that the conditions of our problem can be satisfied only if N t 
tends to the limit zero ; and this consideration, as we shall see, restricts us to 
a determinate series of values of/ 



2j(2;+l) 



^^^ . (23) 

2j(2j + l)-^ 



and by successive applications of this we obtain N s in the form of a convergent 
continued fraction 

£ /3 



y 2j(2j + l) (2j+2)(2? + 3) (2? + 4)(2j + 5) 

' 1 W* .I tif % .I £/ 2 , ' 

2j(2j + l) +1 (2j + 2)(2j + 3) +i (2j + 4)(2j + 5) + - 

(24) 

on the present supposition that N j+t tends with increasing k to the limit 0, 
in the manner indicated by (19). In particular, this formula (24) determines 
the value of N t . Now from (15) we must have 

*i-l-§?S. < 25 > 



216] Free Oscillations 337 

when ce 1 -|L- + -— ^- — gf— ~ °> (26) 

4.5 + 6.7 + "' 

which is equivalent to jVj = oo . This equation determines the admissible 
values of/(= <t/2g>). The constants in (11) are then given by 

B 1 = j3A t B z =JST 2 pA, B^NzNzpA,..., (27> 

where A is arbitrary. 

It is easily seen that when j3 is infinitesimal the roots of (26) are given by 

a *£=V/*=n(n + l), (28) 

where n is an even integer; cf. Art. 199. 

One arithmetically remarkable point remains to be noticed. It might 

appear at first sight that when a value of/ has been found from (26) the 

coefficients B 3 , B 5 , B 7 , ... could be found in succession from (15) and (16), or 

by means of the equivalent formula (18). But this would require us to start 

with exactly the right value of / and to observe absolute accurac}' in the 

subsequent stages of the work. The above argument shews, in fact, that any 

other value, differing by however little, if adopted as a starting point for the 

calculation will inevitably lead at length to values of N, which approximate 

to the limit 1 *. 

An approximation to the longest free period may be attempted by the method of 
Art. 205 a. 

Denoting by £, r] the displacements southwards and eastwards, respectively, we have, 
in the notation of the Art. referred to, 



irpha 2 f" (i 2 + i7 2 ) sin 8d8, M'=2ir P ha 2 (* 



(£ 2 + i] 2 ) sin 8 d8, M' = 2irpha 2 \ (gcos 8. rj -rj . £cos 6) sin 8d8, 



V- T =7rgpa 2 f * £ 2 sin 8d8. 



...(29) 



We will assume that as in the case of no rotation the surface elevation is represented 
by a zonal harmonic of the second order. The formulae (3) of Art. 215 then suggests for our 
assumed type 

£ = A sin 8 cos 8 cos at, rj — B sin 8 cos 3 8 sin at, (30) 

which makes 

C= ?-^(£sin0)=--(3cos 2 0-l),icoso-* (31) 

a sin 8 d8 ^ ' a x ' ' 

We find 

P= 7rp ka 2 (^A 2 + ^ E B i ), Q=&,ir pa >ha*AB, R = ±7rgph 2 A 2 (32) 

The equation (10) of Art. 205 a becomes 

(x 2 -6) A 2 + J(3. %xAB + %B 2 x 2 =0, (33) 

where 

x=aal s f{gh\ p = 4o} 2 a 2 /gh ...(34) 

The stationary values of x are then given by 

#2 = 6 + 10 (35) 

* Sir W. Thomson, I.e. ante p. 335. 



1 



338 Tidal Waves [chap, viit 

For example, taking /3 = 5, which would correspond in the case of the earth to a depth 
of 58080 ft., we find 

aa/J(gh) = 2"854, co/a = '3917. 

The latter number gives the period in terms of the sidereal day. Hence in sidereal time 
27r/o- = 9h. 24 m. The true period, as calculated by Hough (see Art. 222) is 9h. 52 m., but 
this allows for the mutual gravitation of the disturbed water, which we have neglected. 

A correction is however easily made. Since we neglect effect of centrifugal force on 
gravity the influence of T in (29) may be disregarded, whilst the value of V is altered in 
the ratio 

l-f^ = '892, 
Po 
where p 1 /p ( = '18) is the ratio of the density of the water to the mean density of the earth 
(see Art. 200). The result is to replace (35) by 

# 2 =5'352 + f/3 ^36) 

For /3 = 5 this gives a period oi y n. 48 m., in close approximation to Hough's value. 

For greater values of /3, i.e. smaller depths of the ocean, or greater speeds of rotation, 
the approximation is less satisfactory, as we should expect from the nature of our assumed 
type. 

217. It is shewn in the Appendix to this Chapter that the tide-generating 
potential, when expanded in simple- harmonic functions of the time, consists 
of terms of three distinct types. 

The first type is such that the equilibrium tide-height would be given by 

£=#'(i-cos 3 0).cos(a-*-fe) (37)* 

The corresponding forced waves are called by Laplace the ' Oscillations of the 
First Species ' ; the}' include the lunar fortnightly and the solar semi-annual 
tides, and generally all the tides of long period. Their characteristic is 
symmetry about the polar axis, and they form accordingly the most important 
case of forced oscillations of the present type. 

If we substitute from (37) in (7), and assume for 

2 ,» ^- and £ 

expressions of the forms (11) and (12), we have, in place of (14), (15), 

B^WH'-PA-O, (38) 

^-(l-f^i + i/S^O, (39) 

whilst (16) and its consequences hold for all the higher coefficients. It may 
be noticed that (39) may be included under the general formula (16), provided 
we write B_x = — 2H'. It appears by the same argument as before that the 
only admissible solution for an ocean covering the globe is the one that makes 
N^ - 0, and that accordingly Nj must have the value given by the continued 
fraction in (24), where / is now prescribed by the frequency of the disturbing 
forces. 

* In strictness, here denotes the geocentric latitude, but the difference between this and the 
geographical latitude may be neglected consistently with the assumptions introduced in Art. 214. 



216-217] Tides of Long Period 339 

In particular, this formula determines the value of N-y . Now 
B 1 = N 1 B_ l = -2N 1 H', 
and the equation (38) then gives 

A = -W-^N l H'; (40) 

in other words, this is the only value of A which is consistent with a zero 
limit of N 5 , and therefore with a finite velocity at the poles. Any other value 
of A, if adopted as a starting point for the calculation of B\, B 3 , B 5 , ... in 
succession, by means of (38), (39), and (16), would lead ultimately to values 
of Nj approximating to the limit 1. Moreover, since absolute accuracy in the 
initial choice of A and in the subsequent computations would be essential to 
avoid this, the only practical method of calculating the coefficients is to use 
the formulae 

B X \W = -2N x , B 3 = N % B lt B 5 = N S B 3) 
or Bt/H' = - 2 A" a , B s /H' = - 2N X N 2 , B 5 /H' = - 2^ AWs, • . . 

(41) 

where the values of A 7 i, N%, N 3 , ... are to be computed from the continued 
fraction (24). It is evident a posteriori that the solution thus obtained will 
satisfy all the conditions of the problem, and that the series (12) will converge 
with great rapidity. The most convenient plan of conducting the calculation 
is to assume a roughly approximate value, suggested by (19), for one of the 
ratios Nj of sufficiently high order, and thence to compute 

N t . lt Jf/-i, ... N*, tfi 
in succession by means of the formula (23). The values of the constants 
A, B 1} Z? 3 , ..., in (12), are then given by (40) and (41). For the tidal elevation 
we find 

m' = - 2AV/9 - (1 -f*N y ) ^ - JJTi (1 -PNz) ^-... 

-\SiNi...N, ,(l.-/»ffi)Ai*- (42) 

In the case of the lunar fortnightly tide, / is the ratio of a sidereal day 
to a lunar month, and is therefore equal to about ^V> or more precisely '0365. 
This makes f 2 = '00133. It is evident that a fairly accurate representation 
of this tide, and a fortiori of the solar semi-annual tide, and of the remaining 
tides of long period, will be obtained by putting /= ; this materially shortens 
the calculations. 

The results will involve the value of /3, =4f(o 2 a 2 /gh. For £ = 40, which 
corresponds to a depth of 7260 feet, we find in this way 

£/#' = -1515 - 1-0000//2+ 1-5153/x 4 - T2120/* 6 + -6063/* 8 - -2076m 10 

+ -0516/i 12 - -0097/u 14 '+ '0018/* 16 - '0002^ 18 , (43)* 

* The coefficients in (43) and (44) differ only slightly from the numerical values obtained by 
Darwin for the case f= -0365. 



340 Tidal Waves [chap, viii 

whence, at the poles (//,= ± 1), 

?=-f^'x-154, 
and, at the equator (/j, = 0), 

?= Jtf'x'455. 
Again, for /3 = 10, or a depth of 29040 feet, we get 

QH' = -2359 - 1 -0000^ 2 + -5898/i 4 - '1623/* 6 

+ -0258^ - -0026/x 10 + -0002^ 12 (44) 

This makes, at the poles, 

?=-§#' x-470, 
and, at the equator, 

f = J£T x -708. 

For /3 = 5, or a depth of 58080 feet, we find 
?/£T = -2723 - 1-0000/x 2 + 3404^ 4 

- -0509/x 6 + -0043/t 8 - 0004ya 10 (45) 

This gives, at the poles, 

and, at the equator, 

f = \H' x -817. 

Since the polar and equatorial values of the equilibrium tide are — f #' 
and \R' , respectively, these results shew that for the depths in question 
the long-period tides are, on the whole, direct^ though the nodal circles will, 
of course, be shifted more or less from the positions assigned by the equi- 
librium theory. It appears, moreover, that, for depths comparable with the 
actual depth of the sea, the tide has less than half the equilibrium value. 
It is easily seen from the form of equation (7) that with increasing depth, 
and consequent diminution of /3, the tide-height will approximate more and 
more closely to the equilibrium value. This tendency is illustrated by the 
above numerical results. 

It is to be remarked that the kinetic theory of the long-period tides was 
passed over by Laplace, under the impression that practically, owing to the 
operation of dissipative forces, they would have the values given by the 
equilibrium theory. He proved, indeed, that the tendency of frictional forces 
must be in this direction, but it has been maintained by Darwin* that in 
the case of the fortnightly tide, at all events, it is doubtful whether the effect 
would be nearly so great as Laplace supposed. We shall return to this point 
later. 

218. When the disturbance is no longer restricted to be symmetrical 
about the polar axis, we must recur to the general equations (1) and (2) of 
Art. 214. We retain, however, the assumptions as to the law of depth and 
the nature of the boundaries introduced in Art. 215. 

* I.e. ante p. 335. 



217-219] Diurnal Tides 341 

If we assume that O, u, v, fall vary as e l ' (<rt+ ** +<) , where s is integral, th 
equations referred to give 

i*u-2tovceQ0=-2-3 h (Z-'E) i iav + 2coucosd=- -^L(f-{), ...(1) 
ad6 x asmO ^ ; v ' 

... 1 (3 (Aw sin 0) . . ) 

with lo-? = ^-^ l-^— 5Z + W M ( 2 ) 

a sin ( 90 J v ' 

Solving for u, v, we find 



a /cos 3f , . 

4m (/ 2 - cos 2 6)\ f dd b 



.(3) 



where we have written 



G) 2 tt 



?-?=£' a-/- y-» W 

as before. 

It appears that in all cases of simple-harmonic oscillation the fluid particles 
describe ellipses having their principal axes along the meridians and parallels 
of latitude, respectively. 

Substituting from (3) in (2) we obtain the differential equation in f ' : 

_3 f Asin0 (d£ ' s_ , \] 

sin (980 !/ 2 -cos 2 0\a0 + / )\ 

~ p-co& d (? COt ° W + ^ C ° Se ° 2 °) + 4ma ^ = " 4w "^' " ' (5) 
219. The case s = 1 includes, as forced oscillations, Laplace's ' Oscillations 
of the Second Species,' where the disturbing potential is a tesseral harmonic 
of the second order ; viz. 

f=iT"sin0cos<9.cosO£ + (£ + e), (1) 

where cr differs not very greatly from co. This includes the lunar and solar 
diurnal tides. 

In the case of a disturbing body whose proper motion could be neglected, 
we should have a = &>, exactly, and therefore / = \ . In the case of the moon, 
the orbital motion is so rapid that the actual period of the principal lunar 
diurnal tide is very appreciably longer than a sidereal day*; but the sup- 
position that/=|- simplifies the formulae so materially that we adopt it in 
the following investigation!. We find that it enables us to calculate the 
forced oscillations when the depth follows the law 

h = (l-qcos 2 d)h , (2) 

where q is any given constant. 

* It is to be remarked, however, that there is an important term in the harmonic development 
of (2 for which <r — w exactly, provided we neglect the changes in the plane of the disturbing body's 
orbit. This period is the same for the sun as for the moon, and the two partial tides thus produced 
combine into what is called the ' luni-solar ' diurnal tide. 

t Taken with very slight alteration from Airy, "Tides and Waves," Arts. 95 ..., and Darwin, 
Encyc. Brit. (9th ed.), xxiii. 359. 



342 Tidal Waves [chap, viii 

Taking an exponential factor g l >*+*+ e ) ) and therefore putting s=l,/= J, 

in Art. 218 (3), and assuming 

£' = Csin(9cos<9, (3) 

G 
we find u = — i<r — , v = a — .cos0 (4) 

m m 

Substituting in the equation of continuity (Art. 318 (2)), we get 

f+?-£g. ■■■; < 5 > 

which is consistent with the law of depth (2), provided 

fl— . * , H" (6) 

1 — 2qh /ma 

1 his gives c = — ■ f ' ; — £ (7) 

One remarkable consequence of this formula is that in the case of uniform 
depth (q = 0) there is no diurnal tide, so far as the rise and fall of the surface 
is concerned. This result was first established (in a different manner) by 
Laplace, who attached great importance to it as shewing that his kinetic 
theory was able to account for the relatively small values of the diurnal tide 
as then (imperfectly) known, in striking contrast to what would be demanded 
by the equilibrium theory. 

But, although with a uniform depth there is no rise and fall, there are 
tidal currents. It appears from (4) that every particle describes an ellipse 
whose major axis is in the direction of the meridian, and of the same length 
in all latitudes. The ratio of the minor to the major axis is cos 0, and so 
varies from 1 at the poles to at the equator, where the motion is wholly 
N. and S. 

220. In the case 5=2, the forced oscillations of most importance are 
where the disturbing potential is a sectorial harmonic of the second order. 
These constitute Laplace's 'Oscillations of the Third Species/ for which 

f = # ,/, sin 2 0.cos(<7*+2(£ + e), (1) 

where a is nearly equal to 2co. This includes the most important of all the 
tidal oscillations, viz. the lunar and solar semi-diurnal tides. 

If the orbital motion of the disturbing body were infinitely slow we should 
have cr— 2o>, and therefore /= 1; for simplicity we follow Laplace in making 
this approximation, although it is a somewhat rough one in the case of the 
principal lunar tide*. 

A solution similar to that of the preceding Art can be obtained for the 
special law of depth f 

h = h o sin 2 (2) 

* There is, however, a 'luni-solar' semi-diurnal tide whose speed is exactly 2w if we neglect 
the changes in the planes of the orbits. Cf. p. 341, first footnote, 
t Cf. Airy and Darwin, 11. cc. 



219-221] Semi-diurnal Tide 343 

Adopting an exponential factor e ifiu>t+24t+e) , and putting therefore / = 1, 5 = 2, 
we find that if we assume 

?' = Csin 2 0, (3) 

the equations (3) of Art. 218 give 

* ff n l a a n 1 + cos2 # / a \ 

u= — Ccot0, v = -=-C r-3— , (4) 

m 2m sin v 

whence, substituting in Art. 218 (2), 

f=— °.Csin 2 (5) 

ma x ' 

Putting f= f' + f, and substituting from (1) and (3), we find 

C=- _ J, R"\ (6) 

1 — 2h /ma 

and therefore 2Vma g (?) 

1 — zho/ma 

For such depths as actually occur in the ocean 2h <ma, and the tide is 
therefore inverted. It may be noticed that the formulae (4) make the velocity 
infinite at the poles, as was to be expected, since the depth there is zero. 

221. For any other law of depth a solution can only be obtained in the 
form of a series. In the case of uniform depth, we find, putting s = 2, / = 1, 
4,malh = in Art. 218(5), 

(1 -^ 2)2 ^ +{/3(1 -^ 2)2 - v - 6 J ?/= -' 8(1 - /a2)2 ^ (8) 

where //, is written for cos 6. In this form the equation is somewhat intract- 
able, since it contains terms of four different dimensions in /jl. It simplifies 
a little, however, if we transform to 

v, =(1-^)*, = sin <9, 
as independent variable; viz. we find 

1,2(1 ~ v2)d ^' v %~^~ 2p2 ~ ^ 4) f ' = ~ ^ = " ^ E " " 6 ' "* (9) 
which is of three different dimensions in v. 

To obtain a solution for the case of an ocean covering the globe, we assume 

£' = B + B 2 v*+BijA+... + BqvV+ (10) 

Substituting in (9), and equating coefficients, we find 

£ = 0, £ 2 = 0, 0.^4 = 0, (11) 

1 6B 6 - 10£ 4 + /3#"'=0, (12) 

and thenceforward 

2j(2j + 6)B 2j+i -2j(2j + Z)B 2H2 + {3B 2j = (13) 

These equations give B Q> B 8} ... B 2 j, ... in succession, in terms of i? 4 , which 
is so far undetermined. It is obvious, however, from the nature of the 



344 Tidal Waves [chap, viii 

problem, that, except for certain special values of h (and therefore of ft), 
which are such that there is a free oscillation of corresponding type (s = 2) 
having the speed 2w, the solution must be unique. We shall see, in fact, 
that unless B& have a certain definite value the solution above indicated will 
make the meridian component (u) of the velocity discontinuous at the 
equator*. 

The argument is in some respects similar to that of Art. 217. If we 
denote by Nj the ratio B^^By of consecutive coefficients, we have, from (13), 

*** 2j+6 2j(2j + 6)iV ( ^> 

from which it appears that, with increasing j t Nj must tend to one or other 
of the limits and 1. More precisely, unless the limit of Nj be zero, the 
limiting form of N j+1 will be 

(2j+3)/(2; + 6),orl-|, 

approximately. The latter is identical with the limiting form of the ratio 
of the coefficients of v 2j and v 2 J~ 2 in the expansion of (1 — v 2 )^. We infer that, 
unless Bi have such a value as to make N m — 0, the terms of the series (10) 
will become ultimately comparable with those of (1 — v 2 )%, so that we may 
write 

£' = £ + (]. -i^Jf, (15) 

where L, M are functions of v which do not vanish for v = 1. Near the 
equator (v = 1) this makes 

£-*(!-/>»£-** < i6 > 

Hence, by Art. 218 (3), u would change from a certain definite value to an 
equal but opposite value as we cross the equator. 

It is therefore essential, for our present purpose, to choose the value of i? 4 
so that N^ = 0. This is effected by the same method as in Art. 217. Writing 
(13) in the form 



"•-m^r <"> 

we see that Nj must be given by the converging continued fraction 

__£ g B 

Ar 2j (2; + 6) (2j + 2) (2; + 8) (gt + 4) (2f + 10) 

'" 2J + 3 2JTTT 2j+7 (18 > 

2j + 6 2j + 8 2J+10 

* In the case of a polar sea bounded by a small circle of latitude whose angular radius is 
<\tt, the value of B A is determined by the condition that w = 0, or d^/dv = 0, at the boundary. 



22i] Semi-diurnal Tide 345 

This holds from j = 2 upwards, but it appears from (12) that it will give also 
the value of Ni (not hitherto defined), provided we use this symbol for B\\H'" . 
We have then 

Finally, writing £=f + f, we obtain 

qH'" = v 2 + N x v* + N^v* + N x N 2 N*v* + (19) 

As in Art. 217, the practical method of conducting the calculation is to 
assume an approximate value for iVj+i, where j is a moderately large number, 
and then to deduce Nj, iV}_i, ... N 2i A 7 i in succession by means of the 
formula (17). 

The above investigation is taken substantially from the very remarkable paper written 
by Kelvin* in vindication of Laplace's treatment of the problem, as given in the 
Mecanique Celeste. In the passage more especially in question, Laplace determines the 
constant J5 4 by means of the continued fraction for iVj, without, it must be allowed, 
giving any adequate justification of the step ; and the soundness of this procedure had 
been disputed by Airyt, and after him by Ferrel J. 

Laplace, unfortunately, was not in the habit of giving specific references, so that few of 
his readers appear to have become acquainted with the original presentment § of the 
kinetic theory, where the solution for the case in question is put in a very convincing, 
though somewhat different, form. Aiming in the first instance at an approximate 
solution by means of & finite series, thus : 

C = B iV * + B 6 ve+... + B 2k+2 v™ + *, (20) 

Laplace remarks j| that in order to satisfy the differential equations, the coefficients would 
have to fulfil the conditions 

\6B 6 -10B i +l3H'" = 0, \ 

40£ 8 -285 6 + /3£ 4 =0, 



,(21) 



(2* -2) (2£ + 4) B 2k + 2 -(2k-2) (2k + 1) B 2k + pB 21c _ 2 = 0,\ 

- 2k (2£ + 3) B 2k + 2 +(3B 2k =0, j 
(3B 2k + 2 = 0,l 
as is seen at once by putting B 2k + 4 = 0, B 2k + Q = 0, ... in the general relation (13). 

We have here k + l equations between k constants. The method followed is to 
determine the constants by means of the first k relations ; we thus obtain an exact 
solution, not of the proposed differential equation (9), but of the equation as modified by 
the addition of a term @B 2k + 2 v' 2k + 6 to the right-hand side. This is equivalent to an 
alteration of the disturbing force, and if we can obtain a solution such that the required 
alteration is very small, we may accept it as an approximate solution of the problem 
in its original form IT. 

* Sir W. Thomson, "On an Alleged Error in Laplace's Theory of the Tides," Phil. Mag. 
(4), 1. 227 (1875) [Papers, iv. 231]. 

f "Tides and Waves," Art. 111. 

X "Tidal Eesearches," U.S. Coast Survey Rep. 1874, p. 154. 

§ "Becherches sur quelques points du systeme du monde," Mem. de V Acad. roy. des Sciences, 
1776 [1779] [Oeuvres, ix. 187...]. 

|| Oeuvres, ix. 218. The notation has been altered. 

II It is remarkable that this argument is of a kind constantly employed by Airy himself in his 
researches on waves. 



346 



Tidal Waves 



[chap. VIII 



Now, taking the first k relations of the system (21) in reverse order, we obtain B 2k + 2 
in terms of 2? 2fc , thence B 2 k in terms of i>2&-i> an ^ so on > until, finally, 2? 4 is expressed in 
terms of H"' ; and it is obvious that if k be large enough the value of 2? 2 fc + 2> ana tne 
consequent adjustment of the disturbing force which is required to make the solution 
exact, will be very small. This will be illustrated presently, after Laplace, by a numerical 
example. 

The process just given is plainly equivalent to the use of the continued fraction (18) 
in the manner already explained, starting with j+l = k, and iy£=/3/2&(2&+3). The 
continued fraction, as such, does not, however, make its appearance in the memoir here 
referred to, but was introduced in the Mecanique Celeste, probably as an after-thought, as a 
condensed expression of the method of computation originally employed. 

The table below gives the numerical values of the coefficients of the 
several powers of v in the formula (19) for ?/#'", in the cases /3 = 40, 20, 10, 
5, 1, which correspond to depths of 7260, 14520, 29040, 58080, 290400 feet, 
respectively*. The last line gives the value of £/H"' for i> = l, i.e. the ratio 
of the amplitude at the equator to its equilibrium-value. At the poles (v = 0), 
the tide has in all cases the equilibrium- value zero. 





/3 = 40 


= 20 


= 10 


= 5 


= 1 


v i 


+ l'OOOO 


f- 1 -oooo 


■t- 1 -oooo 


+ 1-0000 


+ 1-0000 


V* 


+ 20-1862 


-0-2491 


+6-1915 


+ 0-7504 


+0-1062 


V* 


+ 10-1164 


-1-4056 


+ 3-2447 


+ 0-1566 


+ 0-0039 


V 8 


-13-1047 


-0-8594 


+ 0-7234 


+ 0-0157 


+ 0-0001 


„io 


- 15-4488 


-0-2541 


+ 0-0919 


+ 0-0009 




„12 


- 7-4581 


- 0-0462 


+ 0-0076 






V™ 


- 2-1975 


-0-0058 


+ 0-0004 






„16 


- 0-4501 


- 0-0006 








V 8 


- 0-0687 










„20 


- 0-0082 










„ 22 


- 0-0008 










„24 


- 0-0001 












- 7-434 


-1-821 


+ 11-259 


+ 1-924 


+ 1-110 



We may use the above numerical results to estimate the closeness of the approxi- 
mation in each case. For example, when /3 = 40, Laplace finds B^= — •000004Z/'"' ; the 
addition to the disturbing force which is necessary to make the solution exact would then 
be - •00002/T"i/ 30 , and would therefore bear to the actual force the ratio - '0C002I/ 28 . 

It appears from (19) that near the poles, where v is small, the tides are 
in all cases direct. For sufficiently great depths, /5 will be very small, and 
the formulae (17) and (19) then shew that the tide has everywhere sensibly 
the equilibrium-value, all the coefficients being small except the first, which 
is unity. As h is diminished, /3 increases, and the formula (17) shews that 
each of the ratios iV} will continually increase, except when it changes sign 

* The first three cases were calculated by Laplace, I.e. ante p. 330 ; the last by Kelvin. The 
numbers relating to the third case have been slightly corrected, in accordance with the computa- 
tions of Hough ; see p. 347. 



221-222] Hough's Theory 347 

from + to — by passing through the value oo . No singularity in the 
solution attends this passage of A 7 } through oo , except in the case of N% t 
since, as is easily seen, the product Nj^Nj remains finite, and the coefficients 
in (19) are therefore all finite. But when jVi=oo, the expression for f 
becomes infinite, shewing that the depth has then one of the critical values 
already referred to. 

The table on p. 346 indicates that for depths of 29040 feet, and 
upwards, the tides are everywhere direct, but that there is some critical 
depth between 29040 feet and 14520 feet, for which the tide at the equator 
changes from direct to inverted. The largeness of the second coefficient in 
the case /3 = 40 indicates that the depth could not be reduced much below 
7260 feet before reaching a second critical value. 

Whenever the equatorial tide is inverted, there must be one or more pairs 
of nodal circles (f=0), symmetrically situated on opposite sides of the 
equator. In the case of /3 = 40, the position of the nodal circles is given by 
v = '95, or 6 = 90° ± 18°, approximately*. 

222. The dynamical theory of the tides, in the case of an ocean covering 
the globe, with depth uniform along each parallel of latitude, has been greatly 
improved and developed by Hough f, who, taking up an abandoned attempt 
of Laplace, substituted expansions in spherical harmonics for the series of 
powers of /z, (or v). This has the advantage of more rapid convergence, 
especially, as might be expected, in cases where the influence of the rotation 
is relatively small; and it also enables us to take account of the mutual 
attraction of the particles of water, which, as we have seen in the simpler 
problem of Art. 200, is by no means insignificant. 

If the surface-elevation f, and the conventional equilibrium tide-height f 
(in which the effect of mutual attraction is not included), be expanded in 
series of spherical harmonics, thus 

t-sc, g-s?. (i) 

the complete expression for the disturbing potential will be 

cf. Art. 200. The series on the right hand is to be substituted for f in the 
equations of Arts. 214...; this will be allowed for if we write 

r = SKf,-f n ), (2) 

o 

where a n = l - -£ , (3) 

zn + 1 po 

in modification of the notation of Art. 215 (5) or Art. 218 (4). 

* For a fuller discussion of these points reference may be made to the original investigation 
of Laplace, and to Kelvin's papers. 

f "On the Application of Harmonic Analysis to the Dynamical Theory of the Tides," Phil. 
Trans. A, clxxxix. 201, and cxci. 139 (1897). See also Darwin's Papers, i. 349. 



348 Tidal Waves [chap, viii 

In the oscillations of the 'First Species,' the differential equation may be 
written 

«(£$©+«-• <•» 

If we assume 

S=XC n P n (fi,), t-27„P n (/*), (5) 

we have ?' = 2 (a» C» - 7n) Pn 00 (6) 

Substituting in (4), and integrating between the limits — 1 and /jl, we find 

2 (a. G n - 7n ) (1 - ft *jg + 2/3C„ {(/ 2 - 1) + (1 - ?*)} £*.*• = 0. . . .(7) 

Now, by known formulae of zonal harmonics*, 



/> 1 

and J P n cfy* = 2y| +1 (Pn+i - iVi) 



! /^Pn+2 _ CLP^\ _ 1 (dPn _ dP 



n-2 \ 
U ) 



2/i + l [2n + 3 \ c^yLt rfyit / 2n — 1 \ a(/-t rf/* 

_1 ^Pn+2 2 rfP„ 

(2w + l)(2»+3) rf/i (2n-l)(2» + 3) d/* 

j 1 ^Pn-2 /qx 

(2w-l)(2» + l) (fy* ' '" W 

Substituting in (7), and equating to zero the coefficient of (1 — fx 2 ) -=-^ 
we find 

Cn+t-LnCn+r - g^g ~C n _ 2 =^, ...(10) 



(2rc + 3)(2n + 5) n+ ' n n (27i-3)(2n-l)^- 2 ~y3 ' 

Where X - = ^^) + (2n-l) 2 (2, + 3) -? (U) 

The relation (10) will hold from n = 1 onwards, provided we put 

C-i=3, C = 0. 
The further theory is based substantially on the argument of Laplace, 
given in Art. 221; and the work follows much the same lines as in Arts. 216, 
217, 221. 

In the free oscillations we have y n = 0, and the admissible values of f 
are determined by the transcendental equation 

_1 1 

5.7*.99.11M3 

X2 ~"Lr^"x 6 -&c. -°' (12) 

1 1 



T 3.5 2 .77.9 2 .11 A /1QX 

^-^^L^r^ (13) 

* See Todhunter, Functions of Laplace, &c. c. v. ; Whittaker and Watson, Modern Analysis, 
p. 306. 



222-223] 



Case of Symmetry ; Free Periods 



349 



according as the mode is symmetrical or asymmetrical with respect to the 
equator. Alternative forms of the period equations are given by Hough, 
suitable for computation of the higher roots, and it is shewn that close 
approximations are given by the equations L n = or 

3 p\ gh 2 



= l+n(n + l) 1 



(2ft-l)(2n + 3) 



...(14) 



4o) 2 * ' '" v " ' *"' IV* 2n+ 1 p /4<a> 2 a 2 

except for the first two or three values of n * 

The following table gives the periods (in sidereal time) of the slowest symmetrical 
oscillation (i.e. the one in which the surface-elevation would vary as Pi(ii) if there were 
no rotation), corresponding to various depths f. 

F 





Depth 


0* 


Period 


Period 


p 


(feet) 


4w2 


h. m. 


when u = 
h. m. 


40 


7260 


•44155 


18 3-5 


32 49 


20 


14520 


•62473 


15 11-0 


23 12 


io ! 


29040 


•92506 


12 28-6 


16 25 


5 1 

1 


58080 


1-4785 


9 52-1 


11 35 



The results obtained for the forced oscillations of the ' First Species 5 are very similar 
to those of Art. 217. The limiting form of the long-period tides when o-=0 shews the 
following results : 



i 

1 

p 1 


p/p =-181 


p/p = 


Pole 


Equator 


Pole 


Equator 


40 


•140 


•426 


•154 


•455 


20 


•266 


•551 






10 


•443 


•681 


•470 


•708 


5 


•628 


•796 


•651 


•817 



The second and third columns give the ratio of the polar and equatorial tides to the 
respective equilibrium- values]:. The numbers in the fourth and fifth columns are repeated 
from Art. 217. The comparison shews the effect of the mutual gravitation of the water 
in reducing the amplitude. 

223. In the more general case, where symmetry about the axis is not 
imposed, the surface-elevation f is expanded by Hough in a series of tesseral 
harmonics of the type 

P n s (rie^ t+8 + + * (1) 

* Eeference may also be made to Poole, Proc. Lond. Math. Soc. (2) xix. 299. 

+ The slowest asymmetrical mode has a much longer period. It involves a displacement of 
the centre of mass of the water, so that a correction would be necessary if the nucleus were free ; 
cf. Art. 199. 

X The numbers are deduced from Hough's results. The paper referred to contains discussions 
of other interesting points, including an examination of cases of varying depth, with numerical 
illustrations. 



350 



Tidal Waves 



[chap, vm 



In relation to tidal theory the most important cases are where the disturbing 
potential is of the form (1), with n = 2 and 5=1 or s = 2. 

The calculations are necessarily somewhat intricate*, and it may suffice 
here to mention a few of the more interesting results, which will indicate 
how the gaps in the previous investigations have been filled. 

To understand the nature of the free oscillations, it is best to begin with 
the case of no rotation (« = 0). As co is increased, the pairs of numerically 
equal, but oppositely signed, values of a which were obtained in Art. 199 
begin to diverge in absolute value, that being the greater which has the 
same sign with g>. The character of the fundamental modes is also gradually 
altered. These oscillations are distinguished as ' of the First Class.' 

At the same time certain steady motions which are possible, without 
change of level, when there is no rotation, are converted into long-period 
oscillations with change of level, the speeds being initially comparable with 
&>. The corresponding modes are designated as 'of the Second Class 'f ; 
cf. Art. 206. 

The following table gives the speeds of those modes of the First Class which are of 
most importance in relation to the diurnal and semi-diurnal tides, respectively, and the 
corresponding periods, in sidereal time. The last column repeats the corresponding periods 
in the case of no rotation, as calculated from the formula (15) of Art. 200. 





Second Species 

[8 = 1] 


Third Species 
= 2] 




Depth 
(feet) 


0} 


Period 
h. m. 


(a 


Period 
h. m. 


Period 
when w = 

h. m. 


7260 
14520 
29040 
58080 


1-6337 

- 0-9834 

1-8677 
-1-2450 

2-1641 
-1-6170 

2-6288 
-2-1611 


14 41 
24 24 

12 51 
19 16 

11 5 
14 50 

9 8 
11 6 


1-3347 

-0-6221 

1-6133 

-0-8922 

1-9968 
-1-2855 

2-5535 
-1-8575 


17 59 
38 34 

14 52 
26 54 

12 1 

18 40 

9 24 
12 55 


[ 32 49 
I 23 12 
I 16 25 
I 11 35 



The quickest oscillation of the Second Class has in each case a period of over a day ; 
and the periods of the remainder are very much longer. 

* A simplification is made by Love, "Notes on the Dynamical Theory of the Tides," Proc. 
Lond. Math. Soc. (2) xii. 309 (1913). He writes 

dx d\p dx 5^ 



add a sin 



>d<f> add 



cf. Art. 154 (1). The values of x> ^ are expanded in series of spherical harmonics. 

| These two classes of oscillations have been already encountered in the plane problem of 
Art. 212. 



223] Diurnal and Semi-diurnal Tides 351 

As regards the forced oscillations of the ' Second Species,' Laplace's 
conclusion that when a — a, exactly, the diurnal tide vanishes in the case of 
uniform depth, still holds. The computation for the most important lunar 
diurnal tide, for which ajw — '92700, shews that with such depths as we have 
considered the tides are small compared with the equilibrium heights, and 
are in the main inverted. 

Of the forced oscillations of the 'Third Species,' we may note first the 
case of the solar semi-diurnal tide, for which cr= 2co with sufficient accuracy. 
For the four depths given in our tables, the ratio of the dynamical tide-height 
to the conventional equilibrium tide-height at the equator is found to be 

+ 7-9548, -1-5016, -234-87, +2*1389, 
respectively. 

" The very large coefficients which appear when hg/4xo 2 a 2 = -^ indicate 
that for this depth there is a period of free oscillation of semi-diurnal type 
whose period differs but slightly from half-a-day. On reference to the 
tables ... it will be seen that we have, in fact, evaluated this period as 
12 hours 1 minute, while for the case %/4o> 2 a 2 = -^ we have found a period 
of 12 hours 5 minutes*. We see then that though, when the period of 
forced oscillation differs from that of one of the types of free oscillation by as 
little as one minute, the forced tide may be nearly 250 times as great as the 
corresponding equilibrium tide, a difference of 5 minutes between these 
periods will be sufficient to reduce the tide to less than ten times the 
corresponding equilibrium tide. It seems then that the tides will not tend 
to become excessively large unless there is very close agreement with the 
period of one of the free oscillations. 

" The critical depths for which the forced tides here treated of become 
infinite are those for which a period of free oscillation coincides exactly with 
12 hours. They may be ascertained by putting [a = 2w] in the period- 
equation for the free oscillations and treating this equation as an equation 
for the determination of h The two largest roots are..., and the corre- 
sponding critical depths are about 28,182 feet and 7375 feet 

"It will be seen that in three cases out of the four here considered the effect 
of the mutual gravitation of the waters is to increase the ratio of the tide to 
the equilibrium tide [cf. Art. 221]. In two of the cases the sign is also re- 
versed. This of course results from the fact that whereas when [p/pi = 0*18093] 
one of the periods of free oscillation is rather greater than 12 hours, when 
[p/pi = 0] the corresponding period will be less than 12 hours f." 

Hough has also computed the lunar semi-diurnal tides for which 

£-= 0*96350. 
2(0 

* [Belonging to a mode which comes next in sequence to the one having a period of 17 h. 59 m.] 
f Hough, Phil. Trans. A, cxci. 178, 179. 



352 Tidal Waves [chap, viii 

For the four depths aforesaid the ratios of the equatorial tide-heights to their 
equilibrium-values are found to be 

-2-4187, -1-8000, +110725, +1*9225, 
respectively. 

"On comparison of these numbers with those obtained for the solar 
tides..., we see that for a depth of 7260 feet the solar tides will be direct 
while the lunar tides will be inverted, the opposite being the case when the 
depth is 29,040 feet. This is of course due to the fact that in each of these 
cases there is a period of free oscillation intermediate between twelve solar 
(or, more strictly, sidereal) hours and twelve lunar hours. The critical depths 
for which the lunar tides become infinite are found to be 26,044 feet and 
6448 feet. 

"Consequently this phenomenon will occur if the depth of the ocean be 
between 29,182 feet and 26,044 feet, or between 7375 feet and 6448 feet. 
An important consequence would be that for depths lying between these 
limits the usual phenomena of spring and neap tides would be reversed, the 
higher tides occurring when the moon is in quadrature, and the lower at new 
and full moon* " 

223 a. Some important contributions to the dynamical theory have been 
made by Goldsbrough. Considering, first, the tides in an ocean of uniform depth 
bounded by one or two parallels of latitude, he finds, in the case of a polar 
basin of angular radius 30°, for instance, that for such depths as have been 
considered in Arts. 217, 221 the long-period tides and the semi-diurnal tides do 
not deviate very widely from the values given by the equilibrium theory, when 
this is corrected as explained in the Appendix f. The case is different with 
the diurnal tides, which vary considerably with the size of the basin and the 
depth, and are as a rule considerable, whereas we have seen that in a uniform 
ocean covering the globe they are negligible. 

In the case of an equatorial belt{, the long-period tides again approximate 
to the equilibrium values, whilst the diurnal and semi-diurnal deviate widely, 
to an extent which varies considerably with the latitudes of the boundaries. 

The variations here met with are doubtless conditioned by the relation 
between the imposed period and the natural periods of free oscillation. This 
question has been examined by Goldsbrough with reference to the semi-diurnal 
tides of the Atlantic ocean, which forms a more or less limited and isolated 
system. Taking the case of an ocean limited by two meridians 60° apart, and 
assuming the law of depth 

h = h sin 2 6, 

* Hough, I.e., where reference is made to Kelvin's Popular Lectures and Addresses, London, 
1894, ii. 22 (1868). 

t Proc. Lond. Math. Soc. (2) xiv. 31 (1913). J Ibid. xiv. 207 (1914). 



223-224] Semi-diurnal Tide 353 

he finds* that there will be a free oscillation with <r = 2&> exactly, provided 
ho = 23,200 ft., which means a mean depth of 15,500 ft. With ho = 25,320 ft., 
or a mean depth of 16,880 ft., he finds that the forced tides of the above period 
are still very large compared with the equilibrium values. 

In a more recent paper f by Goldsbrough and Colborne the depth is taken 
to be uniform and equal to the estimated mean depth (12,700 ft.) of the 
Atlantic. For the imposed frequency they take that of the principal semi- 
diurnal constituent (usually denoted by M 2 ) of the lunar disturbing force 
(cr/2o> = *9625). The amplitudes, though not so great as before, prove to be 
largely in excess of the equilibrium values. The diurnal tide in an ocean of 
this type has been investigated by Colborne j. 

224. It is not easy to estimate, in any but the most general way, the 
extent to which the foregoing conclusions of the dynamical theory would 
have to be modified if account could be taken of the actual configuration of 
the ocean, with its irregular boundaries and irregular variation of depth§. 
One or two points may however be noticed. 

In the first place, the formulae (1) of Art. 206 would lead us to expect 
for any given tide a phase-difference, variable from place to place, between 
the tide-height and the disturbing force ||. Thus, in the case of the lunai 
semi-diurnal tides, for example, high-water or low-water need not synchronize 
with the transit of the moon or anti-moon across the meridian. More 
precisely, in the case of a disturbing force of given type for which the 
equilibrium tide-height at a particular place would be 

f = a cos at, (1) 

the dynamical tide-height will be 

£=A cos (at -e), (2) 

where the ratio A /a, and the phase-difference e, will be functions of the 
speed a, as well as of the position of the station. 

Again, consider the superposition of two oscillations of the same type but 
of slightly different speeds, e.g. the lunar and solar semi-diurnal tides. If the 
origin of t be taken at a syzygy, we have 

f = a cos at + a' cos at, (3) 

and £=A cos(at-e) + A' cos (a't-e) (4) 

This may be written 

f = ( A + A' cos <£) cos (at — e) + A' sin </> sin (at - e), (5) 

where = (a - a) t- e + e' (6) 

* Proc. Roy. Soc. A, cxvii. 692 (1927). t Ibid, cxxvi. 1 (1929). 

+ Ibid, cxxxi. 38 (1931). 

§ As to the general mathematical problem reference may be made to Poincare, " Sur l'equi- 
libre et les mouvements des mers," Liouville (5), ii. 57, 217 (1896), and to his Legons de mecanique 
celeste, iii. 

|| This is illustrated by the canal problem of Art. 184. 



354 Tidal Waves [chap, viii 

If the first term in the second member of (4) represents the lunar, and the 
second the solar tide, we shall have a < a, and A > A'. If we write 

A + A' cos <f> = C cos a, A' sin </> = (7 sin a, (7) 

we get f=Ccos(V£-e~a), (8) 

where C = (A* + 2AA' cos d> + A' 2 )*, a = tan" 1 ^' sin * . . . .( 9) 
v T A + J. cos <f> 

This may be described as a simple-harmonic oscillation of slowly varying 
amplitude and phase. The amplitude ranges between the limits A ± A', 
whilst a may be supposed to lie always between ±%ir. The 'speed' must 
also be regarded as variable, viz. we fiad 

d, 4 x <tA 2 + (*+</) A A' cos $ + *' A' 2 „ m 

S^- 41 )- A*+ZAA> G o^Ia>* ( 10 > 

This ranges between 

Aa + A'a , Act — A' a' /,iv* 

A +4' and Z^T (U) 

The above is the well-known explanation of the phenomena of the spring- 
and neap- tides f; but we are now concerned further with the question of 
phase. On the equilibrium theory, the maxima of the amplitude G would 
occur whenever 

{a —a)t = 2n7r, 

where n is integral. On the dynamical theory the corresponding times of 
maximum are given by 

O' - a) *-(€'-€) = 2nir, 
i.e. the dynamical maxima follow the statical by an interval J 

(*' -*)/(</-<,). 

If the difference between a and <r were infinitesimal, this would be equal to 
de/da. 

The fact that the time of high-water, even at syzygy, may follow or 
precede the transit of the moon or anti-moon by an interval of several hours 
is well known §. The interval, when reckoned as a retardation, is, moreover, 
usually greater for the solar than for the lunar semi-diurnal tide, with the 
result that the spring-tides are in many places highest a day or two after 
the corresponding syzygy. The latter circumstance has been ascribed || to 
the operation of Tidal Friction (for which see Chapter xi.), but it is evident 

* Helmholtz, Lehre von den Tonempjindungen (2 9 Aufl.), Braunschweig, 1870, p. 622. 

t Cf . Thomson and Tait, Art. 60. 

X This interval may of course be negative. 

§ The values of the retardations (which we have denoted by e) for the various tidal com- 
ponents, at a number of ports, are given by Baird and Darwin, "Kesults of the Harmonic 
Analysis of Tidal Observations," Proc. R. S. xxxix. 135 (1885), and Darwin, " Second Series of 
Results...," Proc. R. S. xlv. 556 (1889). 

|| Airy, "Tides and Waves," Art. 459. 



224-225] Lag of the Tides 355 

that the phase-differences which are incidental to a complete dynamical 
theory, even in the absence of friction, cannot be ignored in this connection. 
There is reason to believe that they are, indeed, far more important than 
those due to the latter cause. 

Lastly, it was shewn in Arts. 206, 217 that the long-period tides may 
deviate very considerably from the values given by the equilibrium theory, 
owing to the possibility of certain steady motions in the absence of disturbance. 
It has been pointed out by Rayleigh* that these steady motions may be 
impossible in certain cases where the ocean is limited by perpendicular 
barriers. Referring to Art. 214 (6), it appears that if the depth h be 
uniform, f must (in the steady motion) be a function of the co-latitude 6 
only, and therefore by (4) of the same Art., the eastward velocity v must be 
uniform along each parallel of latitude. This is inconsistent with the existence 
of a perpendicular barrier extending along a meridian. The objection would 
not necessarily apply to the case of a sea shelving gradually from the central 
parts to the edgef. 

225. We may complete the investigation of Art. 200 by a brief notice of 
the question of the stability of the ocean, in the case of rotation. 

It has been shewn in Art. 205 that the condition of secular stability is 
that V — T should be a minimum in the equilibrium configuration. If we 
neglect the mutual attraction of the elevated water, the application to the 
present problem is very simple. The excess of the quantity V — T over its 
undisturbed value is evidently 



fj{jV-i«'-*) «**}<*$ W 



where M* denotes the potential of the earth's attraction, 8S is an element of 
the oceanic surface, and the rest of the notation is as before. Since M* - ^co 2 ^ 2 
is constant over the undisturbed level (z = 0), its value at a small altitude z 
may be taken to be gz + const., where, as in Art. 213, 



9 = 



d (v-4«V)l (2) 



dz iz =0 

Since ff£dS = 0, on account of the constancy of volume, we find from (1) that 
the increment of V — T Q is 

iffgPdS (3) 

This is essentially positive, and the equilibrium is therefore 'secularly' stable J. 

* "Note on the Theory of the Fortnightly Tide," Phil. Mag. (6) v. 136 (1903) [Papers, 
iv. 84]. 

f The theory of the limiting forms of long-period tides in oceans of various types is discussed 
by Proudman, Proc. Lond. Math. Soc. (2) xiii. 273 (1913). 

X Cf. Laplace, Mecanique Celeste, Livre 4 me , Arts. 13, 14. 



356 Tidal Waves [chap, vm 

It is to be noticed that this proof does not involve any restriction as to 
the depth of the fluid, or as to smallness of the ellipticity, or even as to 
symmetry of the undisturbed surface with respect to the axis of rotation. 

If we wish to take into account the mutual attraction of the water, the 
problem can only be solved without difficulty when the undisturbed surface 
is nearly spherical, and we neglect the variation of g. The question (as to 
secular stability) is then exactly the same as in the case of no rotation. 
The calculation for this case will find an appropriate place in the next 
chapter (Art. 264). The result, as we might anticipate from Art. 200, is 
that the necessary and sufficient condition of stability of the ocean is that its 
density should be less than the mean density of the earth*. 

226. This is perhaps the most suitable occasion for a few additional 
remarks on the general question of stability of dynamical systems. We 
have in the main followed the ordinary usage which pronounces a state of 
equilibrium, or of steady motion, to be stable or unstable according to the 
character of the solution of the approximate equations of disturbed motion. 
If the solution consists of series of terms of the type Ge ±u } where all the 
values of X are pure imaginary (i.e. of the form ia), the undisturbed state is 
usually reckoned as stable; whilst if any of the Vs are real, it is accounted 
unstable. In the case of disturbed equilibrium, this leads algebraically to 
the usual criterion of a minimum value of V as a necessary and sufficient 
condition of stability. 

It has in recent times been questioned whether this conclusion is, from 
a practical point of view, altogether warranted. It is pointed out that since 
the approximate dynamical equations become less and less accurate as the 
deviation from the equilibrium configuration increases, it is a matter for 
examination how far rigorous conclusions as to the ultimate extent of the 
deviation can be drawn from themf. 

The argument of Dirichlet, which establishes that the occurrence of 
a minimum value of V is a sufficient condition of stability, in any practical 
sense, has already been referred to. No such simple proof is available to 
shew without qualification that this condition is necessary. If, however, we 
recognize the existence of dissipative forces, which are called into play by 
any motion whatever of the system, the conclusion can be drawn as in 
Art. 205. 

A little consideration will shew that a good deal of the obscurity which 
attaches to the question arises from the want of a sufficiently precise 
mathematical definition of what is meant by 'stability.' The difficulty 
is encountered in an aggravated form when we pass to the question of 

* Cf. Laplace, I.e. 

t See papers by Liapounoff and Hadamard, Liouville (5), iii. (1897). 



225-226] Stability of the Ocean 357 

stability of motion. The various definitions which have been propounded 
by different writers are examined critically by Klein and Sommerfeld in 
their book on the theory of the top*. Rejecting previous definitions, they 
base their criterion on the character of the changes produced in the path of 
the system by small arbitrary disturbing impulses. If the undisturbed path 
be the limiting form of the disturbed path when the impulses are indefinitely 
diminished, it is said to be stable, but not otherwise. For instance, the 
vertical fall of a particle under gravity is reckoned as stable, although for 
a given impulsive disturbance, however small, the deviation of the particle's 
position at any time t from the position which it occupied in the original 
motion increases indefinitely with t. Even this criterion, as the writers 
referred to themselves recognize, is not free from ambiguity unless the phrase 
1 limiting form,' as applied to a path, be strictly defined. It appears moreover 
that a definition which is analytically precise may not in all cases be easy to 
reconcile with geometrical prepossessions f. 

The foregoing considerations have reference, of course, to the question 
of 'ordinary' stability. The more important theory of 'secular' stability 
(Art. 205) is not affected. We shall meet with the criterion for this, under 
a somewhat modified form, at a later stage in our subject J. 

* Ueber die Theorie des Kreisels, Leipzig, 1897..., p. 342. 

f Some good illustrations are furnished by Particle Dynamics. Thus a particle moving in a 
circle about a centre of force varying inversely as the cube of the distance will if slightly disturbed 
either fall into the centre, or recede to infinity, after describing in either case a spiral with an 
infinite number of convolutions. Each of these spirals has, analytically, the circle as its 
'limiting form,' although the motion in the latter is most naturally described as unstable. 
Cf. Korteweg, Wiener Ber. May 20, 1886. 

A narrower definition has been given by Love, and applied by Bromwich to several dynamical 
and hydrodynamical problems; see Proc. Lond. Math. Soc. (1) xxxiii. 325 (1901). 

X This summary is taken substantially from the Art. "Dynamics, Analytical," in Encyc. 
Brit. 10th ed. xxvii. 566 (1902), and 11th ed. viii. 756 (1910). 



APPENDIX 

TO CHAPTER VIII 

ON TIDE-GENERATING FORCES 

a. If, in the annexed figure, and C be the centres of the earth and of the disturbing 
body (say the moon), the potential of the moon's attraction at a point P near the earth's 
surface will be — yM/CP, where M denotes the moon's mass, and y the gravitation- 
constant. If we put OC=D, OP=r, and denote the moon's (geocentric) zenith-distance 
at P, viz. the angle POC, by $, this potential is equal to 

yM 

(D 2 -2rDcos$ + r 2 )b' 
P 




We require, however, not the absolute accelerative effect at P, but the acceleration 
relative to the earth. Now the moon produces in the whole mass of the earth an 
acceleration yM/D 2 * parallel to 0(7, and the potential of a uniform field of force of this 
intensity is evidently 

_JL_. rC os& 

Subtracting this from the former result we get, for the potential of the relative attraction 
at % 

Q== yM ^.roosS (1) 

(Z) 2 -*2ri)cos#+r 2 )* ^ 2 

This function Q is identical with the ' disturbing-function ' of planetary theory. 

Expanding in powers of r/D, which is in our case a small quantity, and retaining only 
the most important term, we find 

G=i^ 2 (i-cos^) ( 2 ) 

Considered as a function of the position of P, this is a zonal harmonic of the second 
degree, with OC as axis. 

The reader will easily verify that, to the order of approximation adopted, Q. is equal to 
the joint potential of two masses, each equal to \M, placed, one at (7, and the other at a 
point C in CO produced such that 0C' = 0Cf. 

h. In the 'equilibrium-theory' of the tides it is assumed that the free surface takes 
at each instant the equilibrium-form which might be maintained if the disturbing body 
were to retain unchanged its actual position relative to the rotating earth. In other 

* The effect of this is to produce a monthly inequality in the motion of the earth's centre 
about the sun. The amplitude of the inequality in radius vector is about 3000 miles; that of 
the inequality in longitude is about 7" ; see Laplace, Mecanique Celeste, Livre 6 me , Art. 30, and 
Livre 13 me , Art. 10. 

f Thomson and Tait, Art. 804. These two fictitious bodies are designated as 'moon' and 
'anti-moon,' respectively. 



Equilibrium Theory 359 

words, the free surface is assumed to be a level-surface under the combined action of 
gravity, of centrifugal force, and of the disturbing force. The equation to this level- 
surface is 

*-|o) 2 o; 2 + Q = const., (3) 

where o> is the angular velocity of the rotation, w denotes the distance of any point from 
the earth's axis, and M> is the potential of the earth's attraction. If we use square 
brackets [ ] to distinguish the values of the enclosed quantities at the undisturbed level, 
and denote by £ the elevation of the water above this level due to the disturbing 
potential Q, the above equation is equivalent to 



[¥ - £a> 2 G7 2 ] + f"g- (¥ - |a> 2 OT 2 )l C+ Q =COnst., 



•(4) 



approximately, where dfdz is used to indicate a space-differentiation along the normal 
outwards. The first term is of course constant, and we therefore have 

f--J+4 <*) 



.(6) 



where, as in Art. 213, g— ^- (¥— ^co 2 ar 2 ) 

Evidently, g denotes the value of l apparent gravity ' ; it will of course vary more or less 
with the position of P on the earth's surface. 

It is usual, however, in the theory of the tides, to ignore the slight variations in the 
value of g, and the effect of the ellipticity of the undisturbed level on the surface- value 
of Q. Putting, then, r=a, g = yE/a 2 , where E denotes the earth's mass, and a the mean 
radius of the surface, we have, from (2) and (5), 

C=H (cos* S-D + C, (7) 

where H== ^'~E'\^) ' a > ^ 

as in Art. 180. Hence the equilibrium-form of the free surface is a harmonic spheroid of 
the second order, of the zonal type, whose axis passes through the disturbing body. 

C. Owing to the diurnal rotation, and also to the orbital motion of the disturbing 
body, the position of the tidal spheroid relative to the earth is continually changing, 
so that the level of the water at any particular place will continually rise and fall. 
To analyse the character of these changes, let be the co-latitude, and <f> the longitude, 
measured eastward from some fixed meridian, of any place P, and let A be the north-polar- 
distance, and o the hour-angle west of the same meridian, of the disturbing body. We 
have, then, 

cos£ = cos A cos + sin Asintf cos (a + (f>), (9) 

and thence, by (7), 

C= f H (cos 2 A - 1) (cos 2 6-1) 

+ \H sin 2 A sin 20 cos (a + <£) 

+ \H sin 2 A sin 2 cos 2 (a + <£) + C. (10) 

Each of these terms may be regarded as representing a partial tide, and the results 
superposed. 

Thus, the first term is a zonal harmonic of the second order, and gives a tidal spheroid 
vmmetrical with respect to the earth's axis, having as nodal lines the parallels for which 
cos 2 = ^, or = 90° ±35° 16'. The amount of the tidal elevation in any particular latitude 
varies as cos 2 A — ^. In the case of the moon the chief fluctuation in this quantity has 
a period of about a fortnight; we have here the origin of the 'lunar fortnightly' or 
( declinational ' tide. When the sun is the disturbing body, we have a ' solar semi-annual ' 
tide. It is to be noticed that the mean value of cos 2 A — J with respect to the time is not 



360 Appendix to Chapter VIII 

zero, so that the inclination of the orbit of the disturbing body to the equator involves as 
a consequence a permanent change of mean level. Cf. Art. 183. 

The second term in (10) is a spherical harmonic of the type obtained by putting n = 2, 
s = l in Art. 86 (7). The corresponding tidal spheroid has as nodal lines the meridian 
which is distant 90° from that of the disturbing body, and the equator. The disturbance 
of level is greatest in the meridian of the disturbing body, at distances of 45° N. and S. of 
the equator. The oscillation at any one place goes through its period with the hour- 
angle, a, i.e. in a lunar or solar day. The amplitude is, however, not constant, but varies 
slowly with A, changing sign when the disturbing body crosses the equator. This term 
accounts for the lunar and solar ' diurnal ' tides. 

The third term is a sectorial harmonic (n = 2, -5=2), and gives a tidal spheroid having 
as nodal lines the meridians which are distant 45° E. and W. from that of the disturbing 
body. The oscillation at any one place goes through its period with 2a, i.e. in half a (lunar 
or solar) day, and the amplitude varies as sin 2 A, being greatest when the disturbing body 
is on the equator. We have here the origin of the lunar and solar ' semi-diurnal ' tides. 

The 'constant' C is to be determined by the consideration that, on account of the 
invariability of volume, we must have 

SJ£dS=0, (11) 

where the integration extends over the surface of the ocean. If the ocean cover the 
whole earth we have (7=0, by the general property of spherical surface -harmonics quoted 
in Art. 87. It appears from (7) that the greatest elevation above the undisturbed level is 
then at the points 5=0, 5 = 180°, i.e. at the points where the disturbing body is in 
the zenith or nadir, and the amount of this elevation is §//". The greatest depression is at 
places where 5 = 90°, i.e. the disturbing body is on the horizon, and is ^ff. The greatest 
possible range is therefore equal to IT. 

In the case of a limited ocean, C does not vanish, but has at each instant a definite 
value depending on the position of the disturbing body relative to the earth. This value 
may be easily written down from equations (10) and (11); it is a sum of spherical 
harmonic functions of A, a, of the second order, with constant coefficients in the form of 
surface-integrals whose values depend on the distribution of land and water over the 
globe. The changes in the value of C, due to relative motion of the disturbing body, 
give a general rise and fall of the free surface, with (in the case of the moon) fortnightly, 
diurnal, and semi-diurnal periods. This ' correction to the equilibrium-theory ' as usually 
presented, was first fully investigated by Thomson and Tait* The necessity for a 
correction of the kind, in the case of a limited sea, had however been recognized by 
D. Bernoulli t. 

The correction has an influence on the time of high water, which is no longer synchronous 
with the maximum of the disturbing potential. The interval, moreover, by which high 
water is accelerated or retarded differs from place to place J. 

d. We have up to this point neglected the mutual attraction of the particles of the 
water. To take this into account, we must add to the disturbing potential Q, the 
gravitation-potential of the elevated water. In the case of an ocean covering the earth, 
the correction can be easily applied, as in Art. 200. If we put n = 2 in the formulae of 

* Natural Philosophy, Art. 808; see also Darwin, "On the Correction to the Equilibrium 
Theory of the Tides for the Continents," Proc. Roy. Soc. April 1, 1886 [Papers, i. 328]. It 
appears as the result of a numerical calculation by Prof. H. H. Turner, appended to this paper, 
that with the actual distribution of land and water the correction is of little importance. 

t Traite sur le Flux et Reflux de la Mer, c. xi. (1740). This essay, as well as the one by 
Maclaurin cited on p. 307, and another on the same subject by Euler, is reprinted in Le Seur and 
Jacquier's edition of Newton's Principia. 

% Thomson and Tait, Art. 810. The point is illustrated by the formula (3) of Art. 184 supra. 



Harmonic Analysis 361 

that Art., the addition to the value of Q is — § p/p . gC ; and we thence find without 
difficulty 

?=i4k<»**-» (12) 

It appears that all the tides are increased, in the ratio (1 - fp/p ) _1 - If we assume 
p/p = -18, this ratio is 1*12. 

e. So much for the equilibrium theory. For the purposes of the kinetic theory 
of Arts. 213-224, it is necessary to suppose the value (10) of £ to be expanded in a 
series of simple-harmonic functions of the time. The actual expansion, taking account of 
the variations of A and a, and of the distance D of the disturbing body (which enters 
into the value of H), is a somewhat complicated problem of Physical Astronomy, into 
which we do not enter* 

Disregarding the constant C, which disappears in the dynamical equations (1) of 
Art. 215, the constancy of volume being now secured by the equation of continuity (2), it 
is easily seen that the terms in question will be of three distinct types. 

First, we have the tides of long period, for which 

~£= H' (cos 2 d-%) . cos (o-t + e). .... (13) 

The most important tides of this class are the ' lunar fortnightly ' for which, in degrees 
per mean solar hour, o- = l o, 098, and the 'solar-annual' for which o- = 0°*082. 

Secondly, we have the diurnal tides, for which 

(=H"sm0cosd .cos(o-t + cp + €), (14) 

where <r differs but little from the angular velocity a> of the earth's rotation. These 
include the 'lunar diurnal' (o-=13 0, 943), the 'solar diurnal' (o- = 14°'959), and the 'luni- 
solar diurnal' (o-=<o = 15° '041 ). 

Lastly, we have the semi-diurnal tides, for which 

C=H'"sin 2 0.cos(<Tt + 2<p + e\ (15)t 

where a differs but little from 2o>. These include the 'lunar semi-diurnal' (o- = 28°*984), 
the 'solar semi-diurnal' (<r=30°), and the 'luni-solar semi-diurnal' (<r=2<o=30 0, 082). 

For a complete enumeration of the more important partial tides, and for the values of 
the coefficients H', H", H'" in the several cases, we must refer to the investigations of 
Darwin, already cited. In the Harmonic Analysis of Tidal Observations, which is the 
special object of these investigations, the only result of dynamical theory which is made 
use of is the general principle that the tidal elevation at any place must be equal to the 
sum of a series of simple-harmonic functions of the time, whose periods are the same as 
those of the several terms in the development of the disturbing potential, and are therefore 
known a priori. The amplitudes and phases of the various partial tides, for any particular 
port, are then determined by comparison with tidal observations extending over a 

* Eeference may be made to Laplace, Mecanique Celeste, Livre 13 me , Art. 2. The more 
complete development which has served as the basis of all recent accurate tidal work is due to 
Darwin, and is reprinted in his Papers, i. This development is only quasi-harmonic, certain 
elements which are only slowly variable being treated as constants, but adjustable from time to 
time. A strict harmonic development has recently been carried out by Doodson, Proc. Roy. Soc. 
A, c. 305 (1921). 

f It is evident that over a small area, near the poles, which may be treated as sensibly plane, 
the formulae (14) and (15) make 

fee r cos (<rt + <f> + e) , and f oc r 2 cos (<rt + 2$ + e), 

respectively, where r, w are plane polar co-ordinates. These forms have been used by anticipation 
in Arts. 211, 212. 



362 Appendix to Chapter VIII 

sufficiently long period*. We thus obtain a practically complete expression which can be 
used for the systematic prediction of the tides at the port in question. 

f. One point of special interest in the Harmonic Analysis is the determination of the 
long-period tides. It has been already stated that under the influence of dissipative 
forces these must tend to approximate more or less closely to their equilibrium values. 
In the case of an ocean covering the globe it is at least doubtful whether the dissipative 
forces would be sufficient to produce an appreciable effect in the direction indicated. The 
amplitudes might therefore be expected to fall below those given by the equilibrium theory, 
for the dynamical reason explained in Arts. 206, 214. In the actual ocean, on the other 
hand, this consideration does not apply t, whilst the influence of friction is much greater. 
We may assume, then, that if the earth were absolutely rigid the long-period tides would 
have their full equilibrium values. As a matter of fact the lunar fortnightly, which is 
the only one whose amplitude can be inferred with any certainty from the observations, 
appears to fall short by about one-third. The discrepancy is attributed to elastic yielding 
of the solid body of the earth to the tidal distorting forces exerted by the moon. 

* It is of interest to note, in connection with Art. 187, that the tide-gauges, being situated 
in relatively shallow water, are sensibly affected by certain tides of the second order, which 
therefore have to be taken account of in the general scheme of Harmonic Analysis. 

t See the paper by Eayleigh cited on p. 355 ante. 



CHAPTEK IX 

SURFACE WAVES 

227. We have now to investigate, as far as possible, the laws of wave- 
motion in liquids when the vertical acceleration is no longer neglected. The 
most important case not covered by the preceding theory is that of waves 
on relatively deep water, where, as will be seen, the agitation rapidly 
diminishes in amplitude as we pass downwards from the surface ; but it 
will be understood that there is a continuous transition to the state of things 
investigated in the preceding chapter, where the horizontal motion of the 
fluid was sensibly the same from top to bottom. 

We begin with the oscillations of a horizontal sheet of water, and we will 
confine ourselves in the first instance to cases where the motion is in two 
dimensions, of which one (x) is horizontal, and the other (y) vertical. The 
elevations and depressions of the free surface will then present the appearance 
of a series of parallel straight ridges and furrows, perpendicular to the 
plane xy. 

The motion, being assumed to have been generated originally from rest 
by the action of ordinary forces, will necessarily be irrotational, and the 
velocity-potential <f> will satisfy the equation 

dx* + dy 2 ' w 

with the condition ^- = (2) 

at a fixed boundary. 

To find the condition which must be satisfied at the free surface 
(p = const.), let the origin be taken at the undisturbed level, and let Oy 
be drawn vertically upwards. The motion being assumed to be infinitely 
small, we find, putting 12 = gy in the formula (4) of Art. 20, and neglecting 
the square of the velocity (q), 

J-8r»t>w (3) 

Hence if 77 denote the elevation of the surface at time t above the point (x, 0), 
we shall have, since the pressure there is uniform, 






.(4) 



provided the function F(t), and the additive constant, be supposed merged 
in the value of d<f>/dt. Subject to an error of the order already neglected, 



364 Surface Waves [chap, ix 

this may be written 

<-JEL ; ; <S) 

Since the normal to the free surface makes an infinitely small angle 
(drj/dx) with the vertical, the condition that the normal component of the 
fluid velocity at the free surface must be equal to the normal velocity of the 
surface itself gives, with sufficient approximation, 



.(6) 



drj d(f> 

dt \_dy }, 

This is in fact what the general surface condition (Art. 9 (3)) becomes, if we 
put F(x. y, z y t)=y — r) y and neglect small quantities of the second order. 

Eliminating tj between (5) and (6), we obtain the condition 

dt* +9 dy U ' {) 

to be satisfied when y — 0. This is equivalent to Dp/Dt = 0. 

In the case of simple-harmonic motion, the time-factor being e i{(Tt+e) , this 
condition becomes 

■"+-'! < 8 > 

228. Let us apply this to the free oscillations of a sheet of water, or a 
straight canal, of uniform depth h, and let us suppose for the present that 
there are no limits to the fluid in the direction of x, the fixed boundaries, if 
any, being vertical planes parallel to xy. 

Since the conditions are uniform in respect to x, the simplest supposition 
we can make is that <f> is a simple-harmonic function of x ; the most general 
case consistent with the above assumptions can be derived from this by 
superposition, in virtue of Fourier's Theorem. 

We assume then 

</>=P cos kx.e^ t+t \ (1) 

where P is a function of y only. The equation (1) of Art. 227 gives 

4- ¥P ^' (2) 

whence P = Ae^ + Be~ k y (3) 

The condition of no vertical motion at the bottom is dfyjdy — for y — — h, 
whence 

Ae- kh = Be kh , = %C, say. 

This leads to <£= Ccosh k(y + A,)cos lex .e ii<Tt+e) (4) 

The value of o- is then determined by Art. 227 (8), which gives 

a 2 = gk tanh kh (5) 



227-228] Standing Waves 365 

Substituting from (4) in Art. 227 (5), we find 

7] = — cosh kh cos kx . e i{ot+t) t (6) 

if 

or, writing a = . cosh kh, 

i/ 

and retaining only the real part of the expression, 

7j — a cos kx . sin (at + e) (7) 

This represents a system of 'standing waves,' of wave-length \— 2irjk, 
and vertical amplitude a. The relation between the period (2ir/<r) and the 
wave-length is given by (5). Some numerical examples of this dependence 
are given on p. 369. 

In terms of a we have 

ga cosh k (y + h) . , ^ x /ftX 

^-.^yr 00 ^- 008 ^^) < 8 > 

and it is easily seen from Art. 62 that the corresponding value of the stream- 
function is 

qa sinh k (?/ + h) . , , M x 

If x, y be the co-ordinates of a particle relative to its mean position (x, y), 
we have 

*?-_l!4 <ty__d± nm 

d* 8^' ^ 9y' V ; 

if we neglect the differences between the component velocities at the points 
(#, i/) and (^ + x, y 4- y), as being small quantities of the second order. Sub- 
stituting from (8), and integrating with respect to t, we find 

cosh k(y + h) . 1 . , . , x N 

x = — a . ., . — - sin kx . sin (at + e) 

sinh kh 

$\vfak(y + K) 7 . . . , >. 
y = a . . ,, — - 7 cos kx . sin (at + e), 

sinh kh 

where a slight reduction has been effected by means of (5). The motion of 
each particle is rectilinear, and simple-harmonic, the direction of motion 
varying from vertical, beneath the crests and hollows (kx = m7r), to horizontal, 
beneath the nodes (kx = (m + -|) if). As we pass downwards from the surface 
to the bottom the amplitude of the vertical motion diminishes from <xcos&# 
to 0, whilst that of the horizontal motion diminishes in the ratio cosh kh : 1. 

When the wave-length is very small compared with the depth, kh is large, 
and therefore tanh kh — 1*. The formulae (11) then reduce to 

x = — aepy sin kx . sin (at + e), y = ae ky cos kx . sin (at + e), . . .(12) 

with a 2 = gk (13) 

* This case may of course be more easily investigated independently. 



,(ii) 



366 



Surface Waves 



[chap. IX 



The motion now diminishes rapidly from the surface downwards ; uhus at 
a depth of a wave-length the diminution of amplitude is in the ratio e~ 2n or 
1/535. The forms of the lines of (oscillatory) motion (yjr = const.), for this 
case, are shewn in the annexed figure. 




In the above investigation the fluid is supposed to extend to infinity in 
the direction of x, and there is consequently no restriction to the value of k. 
The formulae also give, however, the longitudinal oscillations in a canal of 
finite length, provided k have the proper values. If the fluid be bounded by 
the vertical planes x = 0, x — l (say), the condition d<j>/dx — is satisfied at 

both ends provided sinkl = 0, or kl^mir, where ra = l, 2, 3, The 

wave-lengths of the normal modes are therefore given by the formula 
X = 2Z/m. Cf. Art. 178. 

229. The investigation of the preceding Art. relates to the case of 
'standing' waves; it naturally claimed the first place, as a straightforward 
application of the usual method of ascertaining the normal modes of oscilla- 
tion of a system about a state of equilibrium. 

In the case, however, of a sheet of water, or a canal, of uniform depth, 
extending horizontally to infinity in both directions, we can, by super- 
position of two systems of standing waves of the same wave-length, obtain 
a system of progressive waves which advance unchanged with constant 
velocity. For this, it is necessary that the crests and troughs of one 
component system should coincide (horizontally) with the nodes of the other, 
that the amplitudes of the two systems should be equal, and that their 
phases should differ by a quarter-period. 

Thus if we put v—Vi±V2> (1) 

where rjx = a sin kx cos at, r) 2 = a cos kx sin at, (2) 

we get 7} = a sin (kx ± at), (3) 

which represents an infinite train of waves travelling in the negative or 
positive direction of x, respectively, with the velocity c given by 

c=^ = (|tanhM)*, (4) 

where the value of a has been substituted from Art. 228 (5). In terms of 



228-229] Progressive Waves 367 

the wave-length (X) we have 

tanh^)* (5) 



V2tt 



X / 

When the wave-length is anything less than double the depth, we have 
tanh kh — 1, sensibly, and therefore* 

'm' ( 6) 



-(DM' 



,2tt; 

On the other hand when X is moderately large compared with h we have 
tanhM = M, nearly, so that the velocity is independent of the wave-length, 
being given by 

c = (gh)i (7) 

as in Art. 170. This result is here obtained on the assumption that the 
wave-profile is a curve of sines, but Fourier's Theorem shews that the 
restriction is now to a great extent unnecessary. 

It appears, on tracing the curve y = (tanh^)/a; > or from a numerical 
table to be given presently, that for a given depth h the wave-velocity 
increases constantly with the wave-length, from zero to the asymptotic 
value (7). 

Let us now fix our attention, for definiteness, on a train of simple-harmonic 
waves travelling in the positive direction, i.e. we take the lower sign in (1) 
and (3). It appears, on comparison with Art. 228 (7), that the value of rji is 
deduced by putting e = \tt, and subtracting \tt from the value of facf, and 
that of 772 by putting e = 0, simply. This proves a statement made above as 
to the relation between the component systems of standing waves, and also 
enables us to write down at once the proper modifications of the remaining 
formulae of the preceding Art. 

Thus, we find, for the component displacements of a particle, 

cosh k (y + h) n \ 

x = Xl -x 2 = a s . nh ^ ' coB(fo-<rQ, 

sinh k(y + h) . /7 jX 

This shews that the motion of each particle is elliptic-harmonic, the period 
(27r/cr, = X/c) being that in which the disturbance travels over a wave-length. 
The semi-axes, horizontal and vertical, of the elliptic orbits are 

cosh k (y + h) . sinh k(y + h) 

a . ^ yT — and a . . , 7 — - , 

smh kh , sinn kh 

respectively. These both diminish from the surface to the bottom {y = — h), 
where the latter vanishes. The distance between the foci is the same for all 

* Green, "Note on the Motion of Waves in Canals," Camb. Trans, vii. (1839) [Papers, 
p. 279]. 

+ This is merely equivalent to a change of the origin from which x is measured. 



.(8) 



368 



Surface Waves 



[chap. IX 



the ellipses, being equal to a cosech kh. It easily appears, on comparison 
of (8) with (3), that a surface-particle is moving in the direction of wave- 
propagation when it is at a crest, and in the opposite direction when it is in 
a trough*. 

When the depth exceeds half a wave-length, e~ kh is very small, and the 
formulae (8) reduce to 

x = ae ky cos (Jcx— o-t), ■y = ae k ysin(k% — <rt), (9) 

so that each particle describes a circle, with constant angular velocity 

a, = (27r<7/\)£-j\ The radii of these circles are given by the formula ae ky , and 

therefore diminish rapidly downwards. 

In the table given below, the second column gives the values of sech kh corresponding 
to various values of the ratio hjX. This quantity measures the ratio of the horizontal 
motion at the bottom to that at the surface. The third column gives the ratio of the 
vertical to the horizontal diameter of the elliptic orbit of a surface-particle. The fourth 
and fifth columns give the ratios of the wave-velocity to that of waves of the same length 
on water of infinite depth, and to that of 'long' waves on water of the actual depth, 
respectively. 

The tables of absolute values of periods and wave-velocities, on the opposite page, are 
abridged from Airy's treatise J. The value of g adopted by him is 32*16 ft./sec. 2 

The possibility of progressive waves advancing with unchanged form is limited, theo- 
retically, to the case of uniform depth ; but the numerical results shew that a variation 
in the depth will have no appreciable influence, provided the depth everywhere exceeds 
(say) half the wave-length. 



h/\ 


sech kh 


tanh kh 


cl(gk-rf 


cl(gh)* 


o-oo 


1-000 


o-ooo 


o-ooo 


1-000 


•01 


•998 


•063 


•250 


•999 


•02 


•992 


•125 


•354 


•997 


•03 


•983 


•186 


•432 


•994 


•04 


•969 


•246 


•496 


•990 


•05 


•953 


•304 


•552 


•984 


•06 


•933 


•360 


•600 


•977 


•07 


•911 


•413 


•643 


•970 


•08 


•886 


•464 


•681 


•961 


•09 


•859 


•512 


•715 


•951 


•10 


•831 


•557 


•746 


•941 


•20 


•527 


•850 


•922 


•823 


•30 


•297 


•955 


•977 


•712 


•40 


•161 


•987 


•993 


•627 


•50 


•086 


•996 


•998 


•563 


•60 


•046 


•999 


•999 


•515 


•70 


•025 


1-000 


1-000 


•477 


•80 


•013 


1-000 


1-000 


•446 


•90 


•007 


1-000 


1-000 


•421 


1-00 


•004 


1-000 


1-000 


•399 


00 


•000 


1-000 


1-000 


•000 



* The results of Arts. 228, 229, for the case of finite depth, were given, substantially, by 
Airy, "Tides and Waves," Arts. 160... (1845). 

t Green, I.e. % "Tides and Waves," Arts. 169, 170. 



229-230] 



Numerical Results 



369 









Length of wave, in feet 


Depth of 








water, 
in feet 


1 


10 


100 j 1000 | 10,000 | 


1 


0*442 


1-873 


17-645 


176-33 


1763-3 




10 


0-442 


1-398 


5-923 


55-80 


557-62 




100 


0-442 


1-398 


4-420 


18-73 


176*45 




1000 


0-442 


1-398 


4-420 


13-98 


59-23 




10,000 


0-442 


1-398 


4-420 


13-98 


44-20 





Depth of 




Length of wave, in feet 




water, 
in feet 


1 


10 100 1000 | 10,000 




1 


2-262 


5-339 


5-667 


5-671 


5-671 


5-671 


10 


2-262 


7-154 


16-88 


17-92 


17-93 


17-93 


100 


2-262 


7-154 


22-62 


53-39 


56-67 


56-71 


1000 


2-262 


7-154 


22-62 


71-54 


168-8 


1.79-3 


10,000 


2-262 


7-154 


22-62 


71-54 


226-2 


567-1 



We remark, finally, that the theory of progressive waves may be obtained, 

without the intermediary of standing waves, by assuming at once, in place of 

Art. 228(1), 

Q = J> e i(<rt-kX) ( 10 ) 

The conditions to be satisfied by P are exactly the same as before, and we 
easily find, in real form, 

7] — a sin (Jex — at), (11) 

qa cosh k (y + h) /7 . __. 

with the same determination of a as before. From (12) all the preceding 
results as to the motion of the individual particles can be inferred without 
difficulty. 

230. The energy of a system of standing waves of the simple-harmonic 
type is easily found. If we imagine two vertical planes to be drawn at unit 
distance apart, parallel to xy, the potential energy per wave-length of the 
fluid between these planes is 

\gp rpdx. 
Jo 

Substituting the value of w from Art. 228 (7), we obtain 

lgpa 2 \. sin 2 (at + e) (1) 



370 Surface Waves [chap, ix 

The kinetic energy is, by the formula (1) of Art. 61, 

dx. 



*'£[+' 



fyjy=o 

Substituting from Art. 228 (8), and remembering the relation between cr and 
k, we obtain 

lgpa 2 \. cos 2 (at + e) (2) 

The total energy, being the sum of (1) and (2), is constant, and equal to 
\gpa 2 \. We may express this by saying that the total energy per unit area 
of the water-surface is \gpci 2 . 

A similar calculation may be made for the case of progressive waves, or 
we may apply the more general argument explained in Art. 174. In either 
way we find that the energy at any instant is half potential and half kinetic, 
and that the total amount, per unit area, is ^gpa 2 . In other words, the 
energy of a progressive wave-system of amplitude a is equal to the work 
which would be required to raise a stratum of the fluid, of thickness a, 
through a height \a. 

231. We next consider the oscillations of the common boundary of two 
superposed liquids which are otherwise unlimited. 

Taking the origin at the mean level of the interface we may write 

<l> = CePy cos Jcaetrt, </>' = C'e-^coskxe™ 1 , (1) 

where the accents relate to the upper fluid. For these satisfy Art. 227 (1) 
atid vanish for y = — oo and y = + oo , respectively. Hence if the equation of 
the disturbed surface is 

7] = a cos kx e i<rt (2) 

we must have 

-kC = kC' = iaa (3) 

by Art. 227 (6). Again, the formulae 

p = K~ m 7 = K- gy (4) 

give p (iaC — ga) = p' (io-C - ga) (5) 

as the condition for continuity of pressure at the interface. Substituting the 
values of G and C" from (3) we have 

•"-«*'^ (6) 

The velocity of propagation of waves of length 27r/& is therefore given by 

C '=f.^: (7) 

fc p + p 

The presence of the upper fluid has therefore the effect of diminishing 
the velocity of propagation of waves of any given length in the ratio 
{(1 —8)/(l +s)}b, where s is the ratio of the density of the upper to that of 



230-231] Superposed Fluids 371 

the lower fluid. This diminution has a two-fold cause; the potential energy 
of a given deformation of the common surface is diminished in the ratio 
1—5, whilst the inertia is increased in the ratio 1 + s *. As a numerical 
example, in the case of water over mercury (s -1 = 13'6) the wave- velocity is 
diminished in the ratio '929. 

It is to be noticed, in this and in other problems of the kind, that there 
is a discontinuity of motion at the common surface. The normal velocity 
(— d(j>/dy) is of course continuous, but the tangential velocity (— d(f>jdoc) changes 
sign as we cross the surface; in other words we have (Art. 151) a vortex-sheet. 
This is an extreme illustration of the remark, made in Art. 17, that the free 
oscillations of a liquid of variable density are not necessarily irrotational. In 
reality the discontinuity, if it could ever be originated, would be immediately 
abolished by viscosity, and the vortex-sheet replaced by a film of vorticityf. 

If p < //, the value of a is imaginary. The undisturbed equilibrium- 
arraugement is then unstable. 

If the two fluids are confined between rigid horizontal planes y = — h y y = h', we assume 
in place of (1) 

$ = C cosh k(y + h) cos kxe i<Tt , <p' = C cosh k(y — h') cos kx e i<rt , (8) 

since these make d<p/dy = 0, dcp'ldy=0 at the respective planes. Hence 

— kCsinh. kh=W sinh kh' = iaa (9) 

The continuity of pressure requires 

p (io~C cosh kh — ga)=p (icrC cosh kh! —go) (10) 

Eliminating C, C, 

g2= gk(p-p') ^ (n) 

p coth&A + p'coth kh' 

When kh and kh' are both very great this reduces to the form (6). When kh' is large and 
kh small we find 

C 2 = .2/ F= A_A^ (12) 

approximately, the main effect of the presence of the upper fluid being the change in the 
potential energy of a given deformation. Its kinetic energy is small compared with that 
of the lower fluid. 

* This explains why the natural periods of oscillation of the common surface of two liquids 
of very nearly equal density are very long compared with those of a free surface of similar extent. 
The fact was noticed by Benjamin Franklin in the case of oil over water; see a letter dated 1762 
{Complete Works, London, n. d., ii. 142). 

Again, near the mouths of some of the Norwegian fiords there is a layer of fresh over salt 
water. Owing to the comparatively small potential energy involved in a given deformation of the 
common boundary, waves of considerable height in this boundary are easily produced. To this 
cause is ascribed the abnormal resistance occasionally experienced by ships in those waters. See 
Ekman, "On Dead- Water," Scientific Results of the Norwegian North Polar Expedition, pt. xv. 
Christiania, 1904. Beference may also be made to a paper by the author, "On Waves due to a 
Travelling Disturbance, with an application to Waves in Superposed Fluids," Phil. Mag. (6), 
xxxi. 386 (1916). 

t The solution, taking account of viscosity, is given by Harrison, Proc. Lond. Math. Soc. (2), 
vi. 396 (1908)- 



372 Surface Waves [chap, ix 

When the upper surface of the upper fluid is free we may assume 

<p <= Ccosh 1c(y + h) cos kx e i(Tt , <p' = (A cosh ky + B sinh ky) cos kx e iat (13) 

The kinematical condition is then 

— kCs'\uhkh-= — B=io-a (14) 

The condition for continuity of pressure at the interface is 

p (io-C cosh kh — ga)=p (iaA—ga) (15) 

The condition for constancy of pressure at the free surface is given by Art. 227 (8) 
provided we put y = h! after the differentiations. Thus 

<r 2 {A cosh kh' + B sinh kh')=gk (A sinh kh! + B cosh kh') (16) 

The elimination of A, B, C between (14), (15), (16) leads to the equation 

o- 4 (p coth kh coth kh' + p) -<x 2 p (coth kh' + coth kh) gk -f (p -p ) g 2 k 2 =0 (17) 

Since this is a quadratic in a 2 , there are two possible systems of waves of any given period 
(27r/o-). This is as we should expect, for when the wave-length is prescribed the system 
has virtually two degrees of freedom, so that there are two independent modes of oscilla- 
tion about the state of equilibrium. For example, in the extreme case where p'/p is small, 
one mode consists mainly in an oscillation of the upper fluid which is almost the same as 
if the lower fluid were solidified, whilst the other mode may be described as an oscillation 
of the lower fluid which is almost the same as if its upper surface were free. 

The ratio of the amplitude at the upper to that at the lower surface is found to be 

fc a , 18) 

kc 2 cosh kh' — g sinh kh! * 

Of the various special cases that may be considered, the most interesting is that in 
which kh is large ; i.e. the depth of the lower fluid is great compared with the wave- 
length. Putting coth kh = l, we see that one root of (17) is now 

* 2 =gk, (19) 

exactly as in the case of a single fluid of infinite depth, and that the ratio of the ampli- 
tudes is e kh '. This is merely a particular case of a general result stated near the end of 
Art. 233 ; it will in fact be found on examination that there is now no slipping at the 
common boundary of the two fluids. 

The second root of (17) is, on the same supposition, 



<T" 



p-p 



p coth kh' + p 
and for this the ratio (18) assumes the value 



fa (20) 



-(t-i\ e -kh' (21) 

If in (20) and (21) we put kh' = cc , we fall back on a former case. If on the other hand 
we make kh' small, we find 

p-HD**- ( 22 > 

and the ratio of the amplitudes is 

(23) 



-a-')- 



These problems were first investigated by Stokes*. The case of any number of super 
posed strata of different densities has been treated by Webbt and Greenhill $ . 

* "On the Theory of Oscillatory Waves," Camb. Trans, viii. (1847) [Papers, i. 212]. 

t Math. Tripos Papers, 1884. 

X "Wave Motion in Hydrodynamics," Amer. Journ. of Math. ix. (1887). 



231-232] Waves on a Surface of Discontinuity 378 

232. Let us next suppose that we have two fluids of densities p, p', one 
beneath the other, moving parallel to x with velocities IT, U', respectively, the 
common surface (when undisturbed) being of course plane and horizontal. 
This is virtually a problem of small oscillations about a state of steady motion. 

We write, then, 

<£ = -£7# + <£j, tf^-U'x + ti, (1) 

where fa, (pi are by hypothesis small. 

The velocity of either fluid at the interface may be regarded as made up 

of the velocity of this surface itself, and the velocity of the fluid relative to it. 

Hence if rj be the ordinate of the displaced surface we have, considering 

vertical components, 

dt^ dx dy' dt* dx ~ dy' w 

as the kinematical conditions to be satisfied for y = 0. 

Again, the formula for the pressure in the lower fluid is 

?-£-*{(*-£)"+(t)'}-»-- 

-%-<%-«+ - <■» 

the terms omitted being either of the second order, or irrelevant to the 
present purpose. Hence the condition of continuity of pressure is 

p(£+*3h*)-'(£ + "'£-") < 4 > 

We have seen, in various connections, that in oscillations about steady 
motion there is not necessarily uniformity of phase throughout the system, 
and in the present case it would not be found possible to satisfy the con- 
ditions on such an assumption. Assuming both fluids to be of unlimited 
depth, the appropriate course is to write 

famCf*****-**, fa' - C'f-*****-**, (5) 

and v **aJ<f*-**> (6) 

The conditions (2) then give 

i(<r-kU)a = -W, i(<r-kU')a = kC' t (7) 

whilst, from (4), 

p{i(* -kU)C '- ga}= p' '{i(tr -leU') C - ga) (8) 

Hence p(<r -kU? + p (a- -rkU'f = gk(p - p) (9) 



or 



.(10) 



T pU+p U' (g p-p pp -p. r> . s 

*- p+p' ± Wp + p ' (p + P 'f (U U) 

The first term on the right-hand side may be called the mean velocity of the 

* These are particular cases of the general boundary-condition (3) of Art. 9, as is seen by 
writing F=y-r), and neglecting small terms of the second order. 



374 Surface Waves [chap, ix 

two currents. Relatively to this there are waves travelling with velocities 
± c, given by 

c2 = c o 2 -r^Y 2 (^- ^') 2 , (ii) 

KP + P) 

where cq denotes the wave- velocity in the absence of currents (Art. 231). It is 
to be noticed however that the values of <r given by (9) are imaginary if 

(U-Uy>{. p2 -^ (12) 

K pp 

The common boundary is therefore unstable for sufficiently small wave- 
lengths. This result would indicate that, if there were no modifying cir- 
cumstances, the slightest breath of wind would ruffle the surface of water. 
A more complete investigation will be given later, taking account of capillary 
forces, which act in the direction of stability. If p = p', or if g — 0, the plane 
form of the surface is (on the present reckoning) unstable for all wave-lengths. 
This result illustrates the statement, as to the instability of surfaces of dis- 
continuity in a liquid, made in Art. 79*. 

The case of p = p\ with U= U', is of some interest, as illustrating the 
flapping of sails and flags f. We may conveniently simplify the question by 
putting U— U' = 0; any common velocity may be superposed afterwards if 
desired. On these suppositions the equation (8) reduces to <r 2 = 0. On 
account of the double root the solution has to be completed by the method 
explained in books on Differential Equations. In this way we obtain the 
two independent solutions 

v = ae ik *, ^ = 0, </>/ = (), (13) 

and v = ate ikx , ^ = -^.e te , fa' = %-^.e ikx (14) 

The former solution represents a state of equilibrium; the latter gives a 
system of stationary waves with amplitude increasing proportionally to the 
time. In this form of the problem there is no physical surface of separation 
to begin with ; but if a slight discontinuity of motion be artificially produced, 
e.g. by impulses applied to a thin membrane which is afterwards dissolved, 
the discontinuity will persist, and, as we have seen, the height of the 
corrugations will continually increase. 

An interesting application of the same method is to the case of a jet of thickness 2b 
moving through still fluid of the same density J. Taking the origin in the medial plane we 
write, for the disturbed jet <£= — Ux + cf) , and for the fluid on the two sides <£ = 0i for 
y > b, and $ = <£ 2 for y < - b. We also denote by r) X , rj 2 the normal displacements of the 
two surfaces y = b and y= — 6, respectively. The proper assumptions are then 

cf> 1 ^A 1 e- k ye i i (Tt - kx \ <t) 2 = A 2 e k ve i ( (Tt - kx ), } 

to-OiW-m, 77 2 =<7 2 e^-**), I (15) 

S = ( A cosh ky + B sinh hy) e<(**-**) . J 

* This instability was first remarked by Helmholtz, I.e. ante p. 22. 

t Rayleigh, Proc. Lond. Math. Soc. (1) x. 4 (1879) [Papers i. 361]. J Rayleigh I.e. 



232-233] Instability of Jets 375 

There are obviously two types of disturbance, in which r} X = r) 2i and rj 1 = -jj 2 , respectively. 
In the former case we have Ci = C 2 , A =0, A 2 = — A x . The kinematical conditions (2) at 
the surface y = b then give 

iaC^kA^-™, i((r-kU)C=-kB coshkh, (16) 

whilst the continuity of pressure requires, gravity being omitted, 

(o- - kU) B $inhkk=<rA 1 e- kh (17) 

Hence 

(o--££0 2 tanhM+o- 2 =0 (18) 

If the thickness 2b is small compared with the wave-length of the disturbance, we have 

<r= ±ikU s l{kh\ (19) 

approximately, indicating a very gradual instability, as is often observed in the case of 
filaments of smoke. 

In the case of symmetry (^ = — rj 2 ), we should find 

(<r-kU) 2 cothkh + <T*=0 (20) 

in place of (18). 



The theory of progressive waves may also be investigated, in a very 
compact manner, by the method of Art. 175*. 

Thus if <f>, -v/r be the velocity- and stream -functions when the problem has 
been reduced to one of steady motion, we assume 

r ?X = _ (# + iy) + i ae ih(x+iy) + ip e -ik(x+iy) } 

C 

whence ® = — x — {air 1 ® — fie 1 ®) sin kx, 

1 a) 

T = - y + (aer 7 ® H- /3e k v) cos ky. 

This represents a motion which is periodic in respect to x, superposed on 
a uniform current of velocity c. We assume that ka and kft are small 
quantities; in other words, that the amplitude of the disturbance is small 
compared with the wave-length. 

The profile of the free surface must be a stream-line; we take it to be 
the line i/r= 0. Its form is then given by (1), viz. to a first approximation 
we have 

y = (a + /3) cos kx, (2) 

shewing that the origin is at the mean level of the surface. Again, at the 
bottom (y= —h) we must also have ty = const. ; this requires 

ae kh + Pe~ kh = 0. 

The equations (1) may therefore be put in the forms 



— = — x + C cosh k(y + h) sin kx, 

c 



* Kayleigh, I.e. ante p. 260. 



y + C sinh k(y + h) cos kx. 
c 



.(3) 



376 Surface Waves [chap, ix 

The formula for the pressure is 

?-— »-»{S)'+d)l 

c 2 
= const. — gy — -= {1 — 2&0 cosh k(y + h) cos &#}, 

if we neglect k 2 C 2 . Since the equation to the stream-line ifr = is 

y = (7 sinh Ich cos A;a?, (4) 

approximately, we have, along this line, 

- = const. 4- (&c 2 coth Ich — g) y. 
P 

The condition for a free surface is therefore satisfied, provided 

9 , tanh hh /KX 

c= ^-^wr" (5) 

This determines the wave-length (2ir/k) of possible stationary undulations on 
a stream of given uniform depth h, and velocity c. It is easily seen that the 
value of hh is real or imaginary according as c is less or greater than (ghft. 

If we impress on everything the velocity — c parallel to x, we get 
progressive waves on still water, and (5) is then the formula for the wave- 
velocity, as in Art. 229. 

When the ratio of the depth to the wave-length is sufficiently great, the 
formulae (1) become 

®== — x + fie 1 ® sin kx, — = — y + fie 1 ® cos kx, (6) 

leading to £ = const. - gy - % {1 - 2k fie 1 ® cos kx + k 2 fi*e 21 ®} (7) 

If we neglect k 2 fi 2 y the latter equation may be written 

- = const. + (kc 2 — g)y + kcyjr (8) 

r 

Hence if c 2 = g/k, (9) 

the pressure will be uniform not only at the upper surface, but along every 
stream-line -ty = const.* This point is of some importance; for it shews that 
the solution expressed by (6) and (9) can be extended to the case of any 
number of liquids of different densities, arranged one over the other in 
horizontal strata, provided the uppermost surface be free, and the total depth 
infinite. And, since there is no limitation to the thinness of the strata, we 
may even include the case of a heterogeneous liquid whose density varies 
continuously with the depth. Cf. Art. 235. 

* This conclusion, it must be noted, is limited to the case of infinite depth. It was first 
remarked by Poisson, I.e. post p. 384. 



.(11) 



233-234] Artifice of Steady Motions 377 

Again, to find the velocity of propagation of waves over the common horizontal 
boundary of two masses of fluid which are otherwise unlimited, we may assume 

^= -y+/3e^cos£#, ^-=-y + 0e-*ycos&F, (10) 

c c 

where the accent relates to the upper fluid. For these satisfy the condition of irrotational 
motion, V 2 ^ = ; and they give a uniform velocity c at a great distance above and below 
the common surface, at which we have \^ = ■*//■', =0, say, and therefore y = /3cos&#, approxi- 
mately. 

The pressure-equations are 

7) (P 1 

*- = const, —gy — ^ (1 - 2k(3e kv cos Tcx\ 
P * 

v' c 2 

S = const. - gy - - (1 + 2^e~ k v cos kx), 

p ' 2, i 

which give, at the common surface, 

— = const. — (g — kc 2 )y, -, = const. — (g+kc 2 )y, (12) 

the usual approximations being made. The condition p=p' thus leads to 

M-^'. ( 13 ) 

as in Art. 231. 

234. As a further example of the method we take the case of two super- 
posed currents, already treated by the direct method in Art. 232. 
The fluids being unlimited vertically, we assume 

yjr= - U{y- fie** cos kx}, yjr' =- U'{y-0er*v coa kx], (1) 

for the lower and upper fluids respectively. The origin is taken at the mean 
level of the common surface, which is assumed to be stationary, and to have 
the form 

y = /3 cos kx (2) 

The pressure-equations give 

- = const. - gy — \ U 2 (1 — 2k fie^ cos kx), 

Pj = const. - gy -%U' 2 (1 + 2k/3er* cos kx), 
P 

whence, at the common surface, 



(3) 



2 = const. + (kU 2 -g)y, ^_ = const. ~(kU' 2 +g)y (4) 

Since we must have p — p f over this surface, we get 

P U* + p'U'*=l(p-p') (5) 

This is the condition for stationary waves on the common surface of the 
two currents U, U'. It may be written 

( pU + p'Uy _ g_ p-l _ 9 p> 

\ p+p' ) k-p+p' (p+p') i(u U)> — w 

which is, easily seen to be equivalent to Art. 232 (10). 



378 Surface Waves [chap, ix 

When the currents are confined by fixed horizontal planes y= — h, y = h', we assume 

(7) 

The condition for stationary waves on the common surface is then found to be 

pU 2 cothkh+p'U ,2 cothkh'=£(p-p') (8)* 

235. The theory of waves in a heterogeneous liquid may be noticed, foi 
the sake of comparison with the case of homogeneity. 

The equilibrium value p of the density will be a function of the vertical co-ordinat( 
(y) only. Hence, writing 

P**Po+P'> P=po+p\ (1) 

where p . is the equilibrium pressure, the equations of motion, viz. 

du dp dv dp 

nr-£> "s--^-«- (2) 

fc+-£+'|-* m 

, du dp' dv dp' , 

become <">Tt = -£' »%--■%-*>> (4) 

!+»lr ' <*> 

small quantities of the second order being omitted. The fluid being incompressible, the 
equation of continuity retains the form 

du dv 

a^ + ^= ' < 6 > 

so that we may write 



■— *• *-s < 7 > 



u— - - "- — 
Eliminating jo' and p' we find t 

fft'«-»S»- ■ 

At a free surface we must have Dp/Dt=0, or 

!—%-•* < 9 > 

Hence, and from (4), we must have 

8^ 9ty 

^ = *aJ ( 10 > 

at such a surface. 

To investigate cases of wave-motion we assume that 

^ oc e i(fft-Jex) *jj\ 

The equation (8) becomes 

3-**£*«-s*)- » 

whilst the condition (10) takes the form 

^"^^ = ° ( 13 ) 

* Greenhill, Z.c. ante p. 372. 

f Cf. Love, "Wave Motion in a Heterogeneous Heavy Liquid," Proc. Lond. Math. Soc. xxii. 
307 (1891). 



■"♦-'&-S)*-° (16) 



234-235] Waves in Heterogeneous Liquid 379 

These are satisfied, whatever the vertical distribution of density, by the assumption 
that \ls varies as e ky . provided 

o*=gk (14) 

For a fluid of infinite depth the relation between wave-length and period is then the same 
as in the case of homogeneity (cf. Art. 229), and the motion is irrotational. 

For further investigations it is necessary to make some assumption as to the relation 
between p and y. The simplest is that 

**«-*, (15) 

in which case (12) takes the form 

w 

The solution is 

^(i^+^e^-^, (17) 

where X 1} X 2 are the roots of 

A*-j8A + (2f-l\iP«0 (18) 

We first apply this to the oscillations of liquid filling a closed rectangular vessel*. 
The quantity k may be any multiple of w/l, where 7 denotes the length. If the equations 
to the horizontal boundaries be y =0, y = h, the condition d\jr/dx=0 gives 

A+B = 0, Ae\ h +Be** h =0, (19) 

whence e(\-^)^ = l, or Xj - X 2 = 2iW/A, (20) 

where 5 is integral. Hence, from (18), # 

X 1 = l i 3-f isir/h, X 2 =J/3-M7r/A, (21) 

and therefore (^-i) ^ 2=x i X 2 = ¥/3 2 + ^ (22) 

We verify that o- is real or imaginary, i.e. the equilibrium arrangement is stable or 
unstable, according as )3 is positive or negative, i.e. according as the density diminishes or 
increases upwards t. 

The case where the fluid (of depth h) has a free surface may serve as an illustration of 
the theory of 'temperature seiches' in lakes J. Assuming the roots of (18) to be complex, say 

X = |/3±ira, (23) 



with m 2 = (^f-l N ) ^ 2 -^, (24) 



2 -i 4 » 



we have y\r = Ce^ v sin my, (25) 

the origin of y being taken at the bottom. The surface-condition (13) gives 



^/3sinraA + mcosmA=^-2 sin mh (26) 

With the help of (24) this may be written 

*»«*-». ggqjpsSiPp . < 27) 

* Kayleigh, "Investigation of the Character of the Equilibrium of an Incompressible Heavy 
Liquid of Variable Density," Proc. Lond. Math. Soc. (1) xiv. 170 [Papers, ii. 200]. Eeference may 
also be made to a paper by the author "On Atmospheric Oscillations," Proc. Boy. Soc. lxxxiv. 
566, 571 (1910), where another law of density is considered. 

t The case of waves on a liquid of finite depth is discussed by Love (I.e.). See also Burnside, 
"On the Small Wave-Motions of a Heterogeneous Fluid under Gravity," Proc. Lond. Math. Soc. 
(1) xx. 392 (1889). 

t Discussed by Wedderburn, Trans. R. S. Edin. xlvii. 619 (1910) and xlviii. 629 (1912). 



380 Surface Waves [chap, ix 

from which the values of mh are to be found. They are given graphically by the inter- 
sections of the curves 

y = tan.z, V=Jj^v (28) 

where n=(3h, a 2 = k 2 h 2 — \$ 2 h 2 . — The only case of interest is when $h is small. We have, 
then, mh = S7r, approximately, and thence 

h 2 h 2 

••-*•??+*»■ (29) 

which is seen to be identical with (22) when the square of /3A is neglected. It appears in 
fact from (25) that the vertical motion at the free surface is very slight. The maximum 
vertical disturbance is at the levels # = (* — £) it. 

When the roots of (18) are real we should get only a slight correction to the formula 
(T 2 =gk tanh hh which holds for a homogeneous fluid. 

236. The investigations of Arts. 227-234 relate to a special type of 
waves; the profile is simple-harmonic, and the train extends to infinity in 
both directions. But since all our equations are linear (so long as we confine 
ourselves to a first approximation), we can, with the help of Fourier's 
Theorem, build up by superposition a solution which shall represent the 
effect of arbitrary initial conditions. Since the subsequent motion is in 
general made up of systems of waves, of all possible lengths, travelling in 
either direction, each with the velocity proper to its own wave-length, the 
form of the free surface will continually alter. The only exception is when 
the wave-length of every system which is present in sensible amplitude is 
large compared with the depth of the fluid. The velocity of propagation, 
viz. \!{gh), is then independent of the wave-length, so that in the case of 
waves travelling in one direction only, the wave-profile remains unchanged 
in form as it advances (Art. 170). 

The effect of a local disturbance of the surface, in the case of infinite 
depth, will be considered presently; but it is convenient to introduce in 
the first place the very important conception of 'group- velocity,' which has 
application, not only to water-waves, but to every case of wave-motion 
where the velocity of propagation of a simple-harmonic train varies with the 
wave-length. 

It has often been noticed that when an isolated group of waves, of sensibly 
the same length, is advancing over relatively deep water, the velocity of the 
group as a whole is less than that of the individual waves composing it. If 
attention be fixed on a particular wave, it is seen to advance through the 
group, gradually dying out as it approaches the front, whilst its former 
place in the group is occupied in succession by other waves which have come 
forward from the rear*. 

The simplest analytical representation of such a group is obtained by the 
superposition of two systems of waves of the same amplitude, and of nearly 

* Scott Kussell, "Keport on Waves," Brit. Ass. Rep. 1844, p. 369. There is an interesting 
letter on this point from W. Froude, printed in Stokes' Scientific Correspondence, Cambridge, 
1907, ii. 156. 



235-236] Group Velocity 381 

but not quite the same wave-length. The corresponding equation of the free 
surface will be of the form 

7) = a sin (kx — <rt) + a sin (k f x— at) 
= 2acos{i(k-k')x-i(<T-<T')t\sm{%(Jc + k')x-l(<T + <T')t}. ...(1) 

If k, k' be very nearly equal, the cosine in this expression varies very slowly 
with x ; so that the wave-profile at any instant has the form of a curve of 
sines in which the amplitude alternates gradually between the values and 
2a. The surface therefore presents the appearance of a series of groups of 
waves, separated at equal intervals by bands of nearly smooth water. The 
motion of each group is then sensibly independent of the presence of the 
others. Since the distance between the centres of two successive groups is 
27r/(A?— k'), and the time occupied by the system in shifting through this 
space is 2ir/(a — a'), the group- velocity ( U, say) is = (a — <r')/(k — k'), or 

*-£. ™ 

ultimately. In terms of the wave-length X (= 27r/k), we have 

. ^r-**. (3) 

where c is the wave -velocity. 

This result holds for any case of waves travelling through a uniform 
medium. In the present application we have 



= (|tanhM) , (4) 

ity, 
, /, 2kh \ 



and therefore, for the group- velocity, 

d(kc )_ , (^ , 2kh 
dk 

The ratio which this bears to the wave-velocity c increases as kh diminishes, 
being \ when the depth is very great, and unity when it is very small, 
compared with the wave-length. 

The above explanation seems to have been first given by Stokes*. The 
extension to a more general type of group was made by Rayleighf and 
GouyJ. 

Another derivation of (3) can be given which is, perhaps, more intuitive. 
In a medium such as we are considering, where the wave-velocity varies with 
the frequency, a limited initial disturbance gives rise in general to a wave- 
system in which the different wave-lengths, travelling with different velocities, 

* Smith's Prize Examination, 1876 [Papers, v. 362]. See also Bayleigh, Theory of Sound, 
Art. 191. 

t Nature, xxv. 52 (1881) [Papers, i. 540]. 

% " Sur la vitesse de la lumiere," Ann. de Chim. et de Phys. xvi. 262 (1889). It has recently 
been pointed out that the theory had been to some extent anticipated by Hamilton, working 
from the optical point of view, in 1839; see Havelock, Cambridge Tracts, No. 17 (1914), p. 6. 



382 



Surface Waves 



[chap. IX 



are gradually sorted out (Arts. 238, 239). If we regard the wave-length \ 
as a function of oc and t, we have 



9X rr^ 

dt doc 



o, 



.(6) 



since X, does not vary in the neighbourhood of a geometrical point travelling 
with velocity U; this is, in fact, the definition of U. Again, if we imagine 
another geometrical point to travel with the waves, we have 



d\ dX _ dc _ dc dX 
dt doc doc dXdoc 



•(7) 



the second member expressing the rate at which two consecutive wave-crests 
are separating from one another. Combining (6) and (7), we are led, again, 
to the formula (3)*. 

This formula admits of a simple geometrical representation t. If a curve be con- 
structed with X as abscissa and c as ordinate, the group-velocity will be represented by 




N 

the intercept made by the tangent on the axis of c. Thus, in the figure, PN represents 
the wave-velocity for the wave-length ON, and OT represents the group-velocity. The 
frequency of vibration, it may be noticed, is represented by the tangent of the angle PON. 

In the case of gravity- waves on deep water, c oc \z ; the curve has the form of the 
parabola y 2 = 4a#, and OT=\PN, i.e. the group-velocity is one-half the wave-velocity. 

237. The group-velocity has moreover a dynamical, as well as a geo- 
metrical, significance. This was first shown by Osborne Reynolds J, in the 
case of deep-water waves, by a calculation of the energy propagated across a 

* See a paper "On Group- Velocity," Proc. Lond. Math. Soc. (2) i. 473 (1904). The subject 
is further discussed by G. Green, "On Group- Velocity, and on the Propagation of Waves in a 
Dispersive Medium," Proc. R. S. Edin. xxix. 445 (1909). 

t Manch. Mem. xliv. No. 6 (1900). 

J "On the Rate of Progression of Groups of Waves, and the Rate at which Energy is 
Transmitted by Waves," Nature, xvi. 343 (1877) [Papers, i. 198]. Reynolds also constructed a 
model which exhibits in a very striking manner the distinction between wave-velocity and group. 
velocity in the case of the transverse oscillations of a row of equal pendulums whose bobs are 
connected by a string. 



236-237] Transmission of Energy 383 

vertical plane. In the case of infinite depth, the velocity-potential corre- 
sponding to a simple-harmonic train 

7) = a sin k (x — ct) (8) 

is <p = ac $ y cos k (x — ct), (9) 

as may be verified by the consideration that for y = we must have 
drj/dt = — d(j)/dy. The variable part of the pressure is p dcf)/dt, if we neglect 
terms of the second order. The rate at which work is being done on the 
fluid to the right of the plane x is therefore 

— I P^r-dy ' — pa 2 k 2 c z sin 2 k (x — ct) I e Uy dy 

J — oo OX J — co 

= \gpa 2 c sin 2 k (x — ct), (10) 

since c 2 = g/k. The mean value of this expression is \gpa 2 c. It appears on 
reference to Art. 230 that this is exactly one-half of the energy of the waves 
which cross the plane in question per unit time. Hence in the case of an 
isolated group the supply of energy is sufficient only if the group advance 
with half the velocity of the individual waves. 

It is readily proved in the same manner that in the case of a finite depth 
h the average energy transmitted per unit time is * 

*^( 1+ bSh)' (11) 

which is, by (5), the same as 

ifl*»'x*g> (12) 

Hence the rate of transmission of energy is equal to the group-velocity, 
d (kc)/dk, found independently by the former line of argument. 

This identification of the kinematical group- velocity of the preceding Art. 
with the rate of transmission of energy may be extended to all kinds of waves. 
It follows indeed from the theory of interference groups (p. 381), which is of 
a general character. For let P be the centre of one of these groups, Q that 
of the quiescent region next in advance of P. In a time r which extends over 
a number of periods, but is short compared with the time of transit of a 
group, the centre of the group will have moved to P', such that PP' = Ur, and 
the space between P and Q will have gained energy to a corresponding 
amount. Another investigation, not involving the notion of 'interference,' was 
given by Rayleigh (I.e.). 

From a physical point of view the group-velocity is perhaps even more 
important and significant than the wave-velocity. The latter may be greater 
or less than the former, and it is even possible to imagine mechanical media 
in which it would have the opposite direction ; i.e. a disturbance might be 

* Rayleigh, "On Progressive Waves," Proc. Lond. Math. Soc. (1) ix. 21 (1877) [Papers, i. 322]; 
Theory of Sound, i. Appendix. 



384 Surface Waves [chap, ix 

propagated outwards from a centre in the form of a group, whilst the in- 
dividual waves composing the group were themselves travelling backwards, 
coming into existence at the front, and dying out as they approach tho rear *. 
Moreover, it may be urged that even in the more familiar phenomena of 
Acoustics and Optics the wave-velocity is of importance chiefly so far as it 
coincides with the group-velocity. When it is necessary to emphasize the 
distinction we may borrow the term 'phase-velocity* from modern Physics to 
denote what is more usually referred to in the present subject as 'wave- 
velocity.' 

238. The theory of the waves produced in deep water by a local dis- 
turbance of the surface was investigated in two classical memoirs by Cauchy f 
and Poisson %. The problem was long regarded as difficult, and even obscure, 
but in its two-dimensional form, at all events, it can be presented in a com- 
paratively simple aspect. 

It appears from Arts. 40, 41 that the initial state of the fluid is deter- 
minate when we know the form of the boundary, and the boundary- values of 
the normal velocity dcf>/dn, or of the velocity-potential <f>. Hence two forms 
of the problem naturally present themselves ; we may start with an initial 
elevation of the free surface, without initial velocity, or we may start with 
the surface undisturbed (and therefore horizontal) and an initial distribution 
of surface-impulse (/o</> ). 

If the origin be in the undisturbed surface, and the axis of y be drawn 
vertically upwards, the typical solution for the case of initial rest is 

7) as cos <rt cos kx, (1) 

<£ =9 e k v coskx, (2) 

provided o- 2 = #&, (3) 

in accordance with the ordinary theory of 'standing waves' of simple- 
harmonic profile (Art. 228). 

If we generalize this by Fourier's double-integral theorem 

f(x) = - dk\ f \a) cos k (x — a) da, (4) 

then, corresponding to the initial conditions 

1 -/(*), 4*> = 0, (5) 

where the zero suffix indicates surface-value (y = 0), we have 

1 f 00 f 00 
V = - I cos atdk\ f (a) cos k (x — a) da, (6) 

"" J J -co" 

Q^lFE^ekyM r f( a )cosk(x-a)da (7) 

* Proc. Lond. Math. Soc. (2) i. 473. f I.e. ante p. 17. 

J "M^moire sur la th^orie des ondes," M€m. de VAcad. Roy. des Sciences, i. (1816). 



237-238] Cauchy-Poisson Wave Problem 385 

If the initial elevation be confined to the immediate neighbourhood of 
the origin, so that /(a) vanishes for all but infinitesimal values of a, we have, 
assuming 

° f(a)da = l, (8) 

— e ky cos kxdk (9) 






This may be expanded in the form 

6 ==^ T \l -~^k + ^k 2 -..\e k y cos kxdk, (10) 

it Jo I o! 5! J 

where use is made of (3). If we write 

— y=r cos 6, x = rsin0, (11) 

we have, y being negative, 



e k ^coshxk n dk-- 1 ^ i Qos(n + l)6, (12)* 



so that (10) becomes 

at (cos 6 1 /t ~ cos 20 1 71 j9 .„cos30 ) /10X 

a result which is easily verified. From this the value of rj is obtained by 
Art. 227 (5), putting = ± |tt. Thus, for x > 0, 

_ 1 \gt 2 1 (gt*s* 1 (gt*\* \ , 

^"tt^^^ "" 3.5V2^/ + 3.5.7.9lW "'} (14JT 

It is evident at once that any particular phase of the surface disturbance, 
e.g., a zero or a maximum or a minimum of 77, is associated with a definite 
value of \gt 2 jx, and therefore that the phase in question travels over the 
surface with a constant acceleration. The meaning of this somewhat remark- 
able result will appear presently (Art. 240). 

The series in (14) is virtually identical with one (usually designated by 
MX) which occurs in the theory of Fresnel's diffraction-integrals. In its 
present form it is convenient only when we are dealing with the initial stages 
of the disturbance; it converges very slowly when \g&\x is no longer small. 
An alternative form may, however, be obtained as follows. 

* This formula may be dispensed with. It is sufficient to calculate the value of at points 
on the vertical axis of symmetry ; its value at other points can then be written down at once by 
a property of harmonic functions (cf. Thomson and Tait, Art. 498). 

t That the effect of a concentrated initial elevation of sectional area Q must be of the form 



V = §f(9t*lx) 



is evident from consideration of ' dimensions. : 
J Cf. Kayleigh, Papers, iii. 129. 



386 Surface Waves [chap, ix 

The surface-value of (f> is, by (9), 

cf>o=-| coakxdk 

7tJ o- 

-j{C^(T + ^) fc "iT™(T" a *)* r } (15) 

Putti ^ '-?(**©• (16) 

we find l" sin (— + at) da =^1°° sin (? -co 2 ) d£, (17) 

[°° sin (--- at\da = 9 - h P° sin (f 2 - a, 2 ) rff , (18) 

wnere a, = (g)* (19) 

Hence ^--^P" sin (f 2 - a> 2 ) d£ (20) 

irx*J o 

From this the value of 77 is derived by Art. 227 (5); thus 

7 7 = ^|r c os(? 2 - G , 2 )^ 
irx*J 

= $-\ jcos co 2 P cos (*ae + sin a> 2 P sin J 2 ^} (21) 

TTX* I JO JO 

This agrees with a result given by Poisson. The definite integrals are 
practically of Fresnel's forms*, and may be considered as known functions. 

Lommel, in his researches on Diffraction f, has given a table of the 
function 

^O + aTTO-"' (22) 

which is involved in (14), for values of z ranging from to 60. We are thus 
enabled to delineate the first nine or ten waves with great ease. The figure 
on the next page shews the variation of 77 with the time, at a particular place ; 
for different places the intervals between assigned phases vary as six, whilst 
the corresponding elevations vary inversely as x. The diagrams on p. 388, on 

* Ju terms of a usual notation we have 

Pcos t*dt=J(fr) C (u), Psin t*di= V^tt) S (u), 

Cu fu 

where C(u)=j cos^-rruPdu, S(w) = | sin ^itu' 2 du, 

Jo Jo 

the upper limit of integration being u = ,J(2lir) . w. Tables of C (u) and S (u) computed by Gilbert 
and others are given in most books on Physical Optics. More extensive tables, due to Lommel, 
are reproduced by Watson, Theory of Bessel Functions, pp. 744, 745. 

t "Die Beugungserscheinungen geradlinig begrenzter Schirme," Abh. d. k. Bayer. Akad. d. 
Wiss. 2° CI. xv. (1886). 



238—239] 



Waves due to a Local Elevation 



387 



the other hand, shew the wave-profile at a particular instant; at different 
times, the horizontal distances between corresponding points vary as the 
square of the time that has elapsed since the beginning of the disturbance, 
whilst corresponding elevations vary inversely as the square of this time. 

1 




[The unit of the horizontal scale is JiVxjg). That of the vertical scale is Qlirx, 
if Q be the sectional area of the initially elevated fluid.] 

When gt 2 l&x is large, we have recourse to the formula (21), which makes 
^-/^-(cosf+sinfY (23) 

approximately, as found by Poisson and Cauchy. This is in virtue of the 
known formulae 



£ cos £*<*?= [J sin ^£=^2 (24) 



Expressions for the remainder are also given by these writers. Thus 
Poisson obtains, substantially, the semi-convergent expansion 
gh < "* 2 



2* 



"Til 



cos v + sin ~) 
4# 4<xJ 



—'•»©'- 



This is derived as follows. We have 



....(25) 



.(26) 



2ia> ' (2i) 2 co 3 (2*) 3 a> 5 ••*' 

by a series of partial integrations. Taking the real part, and substituting in the first line 
of (21), we obtain the formula (25). 

239. In the case of initial impulses applied to the surface, supposed 
undisturbed, the typical solution is 

p(f> = cos at e ky cos Jcx, (27) 

7j = sin cr£ cos for, ,..(28) 

99 



388 



Surface Waves 



[chap. IX 

with a 2 = gk as before. Hence, if the initial conditions be 

p<j>o=F(x), 77 = 0, (29) 

we have </> = — cosate&dkl F (a) cos k (x - a) da, (30) 

irpj o J -co 

rj = <r sin at dk\ F (a) cos k (x — a) da (31) 

-rrgpJo J -oo ' v } 



-100- 




300- 



-400 x 



[The unit of the horizontal scales is \gt\ That of the vertical scales is 2Q/7rgt 2 .] 



239] 



Waves due to a Local Impulse 



389 



For a concentrated impulse acting at the point x = of the surface, we 
have, putting 

00 F(a)da = l, (32) 



/: 

7rpJo 



cos ate 1 ® cos kxdk (33) 



This integral may be treated in the same manner as (9); but it is evident 
that the results may be obtained immediately by performing the operation 
1/gp.d/dt upon those of Art. 238. Thus from (13) and (14) we derive 



v = 



COS ^1.2 C08 ^0 

irp\~r * 9 r 2 

gt M 



t fi 



-px 2 (1 



irpar (i 1.3.5 \2xJ '1.3 
The series in (35) fs related to the function 
z z z z 5 

T7s 



h~M)'--Y <»>• 



.(36) 



1.3.5.7 1.3.5.7.9.11 
which has also been tabulated by Lommel. If we denote the series (22) and 
(36) by Si and 2 8 , respectively, we find 

1 -l^r5 + l.sl7.9 --^ 1+ ^-^ W 

so that the forms of the first few waves can be traced without difficulty. 

The annexed figure shews the rise and fall of the surface at a particular 
7 

60- 
50- 




[The unit of the horizontal scale is J^xjg). That of the vertical scale is 

P / 2 

. / — , where P represents the total initial impulse.] 

* With the help of the theory of ' dimensions ' it is easily seen a priori that the effect of a 
concentrated initial impulse P (per unit breadth) is necessarily of the form 

px 



390 



Surface Waves 



[CHAP. IX 




0-4 



0-5 
"-'a? 



-10000 



0-10 




AP 



[The unit of the horizontal scales is \gt 2 . That of the vertical scales is ^ • 

The upper curve, if continued to the right, would cross the axis of x and would 
thereafter be indistinguishable from it on the present scale.] 



239-240] Interpretation of Results 391 

place; for different places the time-intervals between assigned phases vary as 
\Jx, as in the former case, but the corresponding elevations now vary inversely 
as x%. In the diagrams on the opposite page, which give an instantaneous 
view of the wave-profile, the horizontal distances between corresponding points 
vary as the square of the time, whilst corresponding ordinates vary inversely 
as the cube of the time. 

For large values of \gt 2 jx, we find, performing the operation 1/gp.d/dt 
upon (23), 



^- 6 (cosf-sinf), (38) 

approximately. 



v = o 



240. It remains to examine the meaning and the consequences of the 
results above obtained. It will be sufficient to consider, chiefly, the case of 
Art. 238, where an initial elevation is supposed to be concentrated on a line of 
the surface. 

At any subsequent time t the surface is occupied by a wave-system whose 
advanced portions are delineated on p. 388. For sufficiently small values of 
x the form of the waves is given by (23); hence as we approach the origin 
the waves are found to diminish continually in length, and to increase 
continually in height, in both respects without limit. 

As t increases, the wave-system is stretched out horizontally, proportionally 
to the square of the time, whilst the vertical ordinates are correspondingly 
diminished, in such a way that the area 



Irjdx 



included between the wave-profile, the axis of x, and the ordinates corre- 
sponding to any two assigned phases (i.e. two assigned values of a>) is 
constant*. The latter statement may be verified immediately from the mere 
form of (14) or (21). 

The oscillations of level, on the other hand, at any particular place, are 
represented on p. 387. These follow one another more and more rapidly, with 
ever increasing amplitude. For sufficiently great values of t, the course 
of these oscillations is given by (23). 

In the region where this formula holds, at any assigned epoch, the 
changes in length and height from wave to wave are very gradual, so that 
a considerable number of consecutive waves may be represented approxi- 

* This statement does not apply to the case of an initial impulse. The corresponding pro- 
position then is that 



,dar, 
taken between assigned values of w, is constant. This appears from (34). 



]<Po l 



392 Surface Waves [chap, ix 

mately by a curve of sines. The circumstances are, in fact, all approximately 
reproduced when 

*£"** (»») 

Hence, if we vary t alone, we have, putting At = t, the period of oscillation, 

T =-^- ; ( 4 °) 

whilst, if we vary x alone, putting Ax — — X, where \ is the wave-length, 

we find 

8ttx 2 

9* 

The wave-velocity is to be found from 



X = -3T (41) 



a£ = 0; (42) 

,i • A# 2x /q\ //lox 

this s ives ^=t = \/L' (43 > 

by (41), as in the case of an infinitely long train of simple-harmonic waves 
of length \. 

We can now see something of a reason why each wave should by con- 
tinually accelerated. The waves in front are longer than those behind, and 
are accordingly moving faster. The consequence is that all the waves are 
continually being drawn out in length, so that their velocities of propagation 
continually increase as they advance. But the higher the rank of a wave in 
the sequence, the smaller is its acceleration. 

So far, we have been considering the progress of individual waves. But, 
if we fix our attention on a group of waves, characterized as having (approxi- 
mately) a given wave-length \, the position of this group is regulated 
according to (43) by the formula 

?=*v/fe; ^ 

i.e. the group advances with a constant velocity equal to half that of the 
component waves. The group does not, however, maintain a constant 
amplitude as it proceeds; it is easily seen from (23) that for a given value 
of A, the amplitude varies inversely as sjx. 

It appears that the region in the immediate neighbourhood of the origin 
may be regarded as a kind of source, emitting on each side an endless 
succession of waves of continually increasing amplitude and frequency, whose 
subsequent careers are governed by the laws above explained. This persistent 
activity of the source is not paradoxical; for our assumed initial accumulation 
of a finite volume of elevated fluid on an infinitely narrow base implies an 
unlimited store of energy. 



24o] Interpretation of Results 393 

In any practical case, however, the initial elevation is distributed over 
a band of finite breadth; we will denote this breadth by I. The disturbance 
at any point P is made up of parts due to the various elements, 8a, say, of 
the breadth I; these are to be calculated by the preceding formulae, and 
integrated over the breadth of the band. In the result, the mathematical 
infinity and other perplexing peculiarities, which we meet with in the case 
of a concentrated line-source, disappear. It would be easy to write down the 
requisite formulae, but, as they are not very tractable, and contain nothing 
not implied in the preceding statement, they may be passed over. It is 
more instructive to examine, in a general way, how the previous results will 
be modified. 

The initial stages of the disturbance at a distance x, such that Ijx is 
small, will evidently be much the same as on the former hypothesis; the 
parts due to the various elements 8a will simply reinforce one another, and 
the result will be sufficiently expressed by (14) or (23) provided we multiply by 



/ 



QO 

f(a) da, 

-8 



i.e. by the sectional area of the initially elevated fluid. The formula (23), 
in particular, will hold when \gft\x is large, so long as the wave-length \ 
at the point considered is large compared with I, i.e. by (41), so long as 
\gt 2 jx . Ijx is small. But when, as t increases, the length of the waves at x 
becomes comparable with or smaller than I, the contributions from the 
different parts of I are no longer sensibly in the same phase, and we have 
something analogous to 'interference' in the optical sense. The result 
will, of course, depend on the special character of the initial distribution of 
the values of f{a) over the space I*, but it is plain that the increase of 
amplitude must at length be arrested, and that ultimately we shall have 
a gradual dying out of the disturbance. 

There is one feature generally characteristic of the later stages which 
must be more particularly adverted to, as it has been the cause of some 
perplexity; viz. a fluctuation in the amplitude of the waves. This is readily 
accounted for on 'interference' principles. As a sufficient illustration, let 
us suppose that the initial elevation is uniform over the breadth I, and that 
we are considering a stage of the disturbance so late that the value of A, in 
the neighbourhood of the point x under consideration has become small com- 
pared with I. We shall evidently have a series of groups of waves separated 
by bands of comparatively smooth water, the centres of these bands occurring 
wherever I is an exact multiple of X, say I = ri\. Substituting in (41), we find 



* 2 V 2mr' ^ 



5) 
i.e. the bands in question move forward with a constant velocity, which is, in 

* Cf. Burnside, "On Deep-water Waves resulting from a Limited Original Disturbance," 
Proc. Land. Math. Soc. (1) xx. 22 (1888). 



394 Surface Waves [chap, ix 

fact, the group-velocity corresponding to the average wave-length in the 
neighbourhood*. 

The ideal solution of Art. 238 necessarily fails to give any information as to what 
takes place at the origin itself. To illustrate this point in a special case, we may assume 

M-it/h* (46) 

the formula (7) then gives 



♦-*/. 



°° s iE^ e t(»-b) C0S ^^ (47) 

o a- 



The surface-elevation at the origin is 

r, = Q I™ cos <Tte-* b dk=^( C ° cos at £-<*>!* *d<r = ^ % /"" sin ate- **>/<> da. ...(48) 
ir Jo -ngj o Trgdtjo 

By a known formula we havet 

[* > e-* 2 sm2Pxdx=e-f i2 /"" e* 2 dx (49) 

Hence, putting a> 2 =gt 2 /4:b, (50) 

we find ri = -% 4- • e" w2 fV*P = X(l~ 2<oe~» 2 [" <* 2 dx\ (51) 

irb da> Jo tto\ Jo / 

Hence 4~ (*«**)- -?$■ ["**<&, (52) 

d(o w ' irb J o 

shewing that rje^ steadily diminishes as t increases. Hence r] can only change sign once. 
The form of the integrals in (48) shews that rj tends finally to the limit zero ; and it may 
be proved that the leading term in its asymptotic value is — 2QJirgt 2 X. 

One noteworthy feature in the above problems is that the disturbance is propagated 
instantaneously to all distances from the origin, however great. Analytically, this might be 
accounted for by the fact that we have to deal with a synthesis of waves of all possible 
lengths, and that for infinite lengths the wave-velocity is infinite. It has been shewn, 
however, by Rayleigh § that the instantaneous character is preserved even when the water 
is of finite depth, in which case there is an upper limit to the wave-velocity. The physical 
reason of the peculiarity is that the fluid is treated as incompressible, so that changes of 
pressure are propagated with infinite velocity (cf. Art. 20). When compressibility is taken 
into account a finite, though it may be very short, interval elapses before the disturbance 
manifests itself at any point ||. 

* This fluctuation was first pointed out by Poisson, in the particular case where the initial 
elevation (or rather depression) has a parabolic outline. 

The preceding investigations have an interest extending beyond the present subject, as 
shewing how widely the effects of a single initial impulse in a dispersive medium [i.e. one in 
which wave-velocity varies with wave-length) may differ from what takes place in the case of 
sound, or in the vibrations of an elastic solid. The above discussion is taken, with some modifica- 
tions, from a paper "On Deep- Water Waves," Proc. Lond. Math. Soe. (2) ii. 371 (1904), where 
also the effect of a local periodic pressure is investigated. 

t This formula presents itself as a subsidiary result in the process of evaluating 

I e~ x2 cos 2[3x dx 
by a contour integration. 

% The definite integral in (52) has been tabulated by Dawson, Proc. Lond. Math. Soc. (1) 
xxix. 519 (1898), and the function in (49) by Terazawa, Science Reports of the Univ. of Tokio, 
vi. 171 (1917). 

§ "On the Instantaneous Propagation of Disturbance in a Dispersive Medium, ...," Phil. 
Mag. (6), xviii. 1 (1909) [Papers, v. 514]. See also Pidduck, " On the Propagation of a Disturb- 
ance in a Fluid under Gravity," Proc. Roy. Soc. A, lxxxiii. 347 (1910). 

|| Pidduck, "The Wave-Problem of Cauchy and Poisson for Finite Depth and slightly Com 
pressible Fluid," Proc. Roy. Soc. A, lxxxvi. 396 (1912). 



240-24i] Principle of Stationary Phase 395 

241. The space which has been devoted to the above investigation may 
be justified by its historical interest, and by the consideration that it deals 
with one of the few problems of the kind which can be solved completely. 
It was shewn, however, by Kelvin that an approximate representation of 
the more interesting features can be obtained by a simpler process, which has 
moreover a very general application*. 

The method depends on the approximate evaluation of integrals of the 
type ^ 

u= I $(m)Jf*dx (1) 

J a 

It is assumed that the circular function goes through a large number of 
periods within the range of integration, whilst (f)(x) changes comparatively 
slowly; more precisely it is assumed that, when f(x) changes by 2tt, (f>(x) 
changes by only a small fraction of itself. Under these conditions the various 
elements of the integral will for the most part cancel by annulling interference, 
except in the neighbourhood of those values of x, if any, for which f(x) is 
stationary. If we write x = a + f , where a is a value of x, within the range 
of integration, such that f (a) = 0, we have, for small values of f , 

/(*)=/(«) +W(«0, (2) 

approximately. The important part of the integral, corresponding to values 
of x in the neighbourhood of a, is therefore equal to 

A(a)eW P e^w.P^ (3) 

J -8 

approximately, since, on account of the fluctuation of the integrand, the 
extension of the limits to + 00 causes no appreciable error. Now by a known 
formula (Art. 238 (24)) we have 

T e ^*Pd!; = ^ }±r= —-<>*** W 

J _oo m y/2 m 

Hence (3) becomes 

^ ( a) &m*k* t (5) 



Vli/"(«)| 

where the upper or lower sign is to be taken in the exponential according as 
/" (a) is positive or negative. 

If a coincides with one of the limits of integration in (1), the limits in (3) 
will be replaced by and 00 , or — 00 and 0, and the result (5) is to be halved. 

If the approximation in (2) were continued, the next term would be 
if 8 /'"( a ); the foregoing method is therefore only valid under the condition 

* Sir W. Thomson, "On the Waves produced by a Single Impulse in Water of any Depth, 
or in a Dispersive Medium," Proc. R. S. xlii. 80 (1887) [Papers, iv. 303]. The method of treating 
integrals of the type (1) had however been suggested by Stokes in his paper ' ' On the Numerical 
Calculation of a Class of Definite Integrals and Infinite Series," Camb. Tram. ix. (1850) [Papers, 
ii. 341, footnote]. 



396 Surface Waves [chap, ix 

that %f" (a)//" (a) must be small even when f&f" (a) is a moderate multiple 
of 2tt. This requires that the quotient 

/"' («)/{!/" («)l)* 

should be small. 

Suppose now that, in a medium of any kind, an initial disturbance, whether 
of the nature of impulse or displacement, of amount cos kx per unit length, 
gives rise to an oscillation of the type 

rj = <j>(k) cos kxe i<rt , (6) 

where a is a function of h determined by the theory of free waves. The effect 
of a concentrated unit initial disturbance is then given by the Fourier ex- 
pression 

v = ~ [ °° 6 (k) **<**-*■» dk+^- r<b (k) e«»*+**> dk (7) 

It is understood that in the end only the real part of the expressions is to be 
retained. 

The two terms in (7) represent the result of superposing trains of simple- 
harmonic waves of all possible lengths, travelling in the positive and negative 
directions of x, respectively. If, taking advantage of the symmetry, we confine 
our attention to the region lying to the right of the origin, the exponential in 
the first integral will alone, as a rule*, admit of a stationary value or values, 
viz. when 

*£- < 8 > 

This determines k, and therefore also a, as a function of x and t, and we then 
find, in accordance with (5), 

^ viw%^ r cos( "*~ fa±i,r) ' (9) 

where the ambiguous sign follows that of d 2 <r/dJ<?. The approximation postu- 
lates the smallness of the ratio 

d*a/dk* + *J{t\d*cr/dk 2 \ z } (10) 

Since 






.(11) 



dt 

by (8), it appears that the wave-length and the period in the neighbourhood 
of the point x at time t are 27r/k and 27r/cr, respectively. The relation (8) 
shews that the wave-length is such that the corresponding group-velocity 
(Art. 236) is xjt. 

* If the group-velocity were negative, as in some of the artificial cases referred to in Art. 237, 
the second integral would be the important one. 



24l] 



Graphical Illustrations 



397 



The above process, and the result, may be illustrated by various graphical constructions*. 
The simplest, in some respects, is based on a slight modification of the diagram of Art. 236. 
We construct a curve with X as abscissa and ct as ordinate, where t denotes the time that 
has elapsed since the beginning of the disturbance. To ascertain the nature of the wave- 
system in the neighbourhood of any point x, we measure off a length OQ, equal to x, along 
the axis of ordinates. If PN be the ordinate corresponding to any given abscissa X, the 
phase of the disturbance at x, due to the elementary wave-train whose wave-length is X, 
will be given by the gradient of the line QP ; for if we draw QR parallel to ON, we have 

PR _ PN- OQ = ct-x = *t-kx 

QR~ ON ~ X ~ 2tt 



,(12) 




Hence the phase will be stationary if QP be a tangent to the curve ; and the predominant 
wave-lengths at the point x are accordingly given by the abscissae of the points of contact 
of the several tangents which can be drawn from Q. These are characterized by the property 
that the group-velocity has a given value x/t. 

If we imagine the point Q to travel along the straight line on which it lies, we get an 
indication of the distribution of wave-lengths at the instant t for which the curve has 
been constructed. If we wish to follow the changes which take place in time at a given 
point x, we may either imagine the ordinates to be altered in the ratio of the respective 
times, or we may imagine the point Q to approach in such a way that OQ varies inversely 
as t. 



crt=koc 




* Proc. of the 5th Intern. Congress of Mathematicians, Cambridge, 1912, p. 281. 



398 Surface Waves [chap, ix 

The foregoing construction has the defect that it gives no indication of the relative 
amplitudes in different parts of the wave-system. For this purpose we may construct the 
curve which gives the relation between at as ordinate and k as abscissa. If we draw a 
line through the origin whose gradient is a?, the phase due to a particular elementary wave- 
train, viz. o-t - kx, will be represented by the difference of the ordinates of the curve and 
the straight line. This difference will be stationary when the tangent to the curve is 
parallel to the straight line, i.e. when tdo-\dk — x, as already found. It is further evident 
that the phase-difference, for elementary trains of slightly different wave-lengths, will 
vary ultimately as the square of the increment of k. Also that the range of values of k for 
which the phase is sensibly the same will be greater, and consequently the resulting dis- 
turbance will be more intense, the greater the vertical chord of curvature of the curve. This 
explains the occurrence of the quantity td 2 <r/dk 2 in the denominator of the formula (9). 

In the hydrodynamical problem of Art. 238 we have* 

*(*) = i, °*=gh (13) 

whence 

da\&k = \g±k~\ d 2 <7/dkZ=-lgik-%, d*a/<tt? = $gik-*. ...(14) 

Hence, from (8), 

k = gt 2 /4>x 2 , * = gt/2x, (15) 

and therefore 

9** i 

77 = ^ pUg&lto-t*) 

V(2?r)«* 
or, on rejecting the imaginary part, 

•-&-£-««•) < u > 

The quotient in (10) is found to be comparable with (2x/gt 2 )%, so that the 
approximation holds only for times and places such that \gt 2 is large com- 
pared with x. 

These results are in agreement with the more complete investigation of 
Art. 238. The case of Art. 239 can be treated in a similar manner. 

It appears from (16), or from the above geometrical construction (the 
curve being now a parabola as in Art. 236), that in the procession of waves 
at any instant the wave-length diminishes continually from front to rear ; 
and that the waves which pass any assigned point will have their wave-lengths 
continually diminishing f. 

242. We may next calculate the effect of an arbitrary, but steady, 
application of pressure to the surface of a stream. We shall consider only 
the state of steady motion which, under the influence of dissipative forces, 

* The difficulty as to convergence in this case is met by the remark that the formula (9) 
of Art. 238 gives 

7} = - ^5 = lim y -*. - / e k v cos at cos kx dk, 
g ot ttJo 

where y is negative before the limit. 

t For further applications reference may be made to Havelock, ' « The Propagation of Waves 
in Dispersive Media...," Proc. Roy. Soc. lxxxi. 398 (1908). 



241-242] Surface Disturbance of a Stream 399 

however small, will ultimately establish itself*. The question is in the first 
instance treated directly ; a briefer method of obtaining the principal result 
is explained in Art. 248. 

It is to be noted that in the absence of dissipative forces, the problem is to 
a certain extent indeterminate, for we can always superpose an endless train 
of free waves of arbitrary amplitude, and of wave-length such that their 
velocity relative to the water is equal and opposite to that of the stream, in 
which case they will maintain a fixed position in space. 

To avoid this indeterminateness, we may avail ourselves of an artifice 
due to Rayleigh, and assume that the deviation of any particle of the fluid 
from the state of uniform flow is resisted by a force proportional to the 
relative velocity. 

This law of friction does not profess to be altogether a natural one, 
but it serves to represent in a rough way the effect of small dissipative forces ; 
and it has the great mathematical convenience that it does not interfere with 
the irrotational character of the motion. For if we write, in the equations of 
Art. 6, 

X = -/jl(u — c), Y= — g — fjLV, Z= — fiw, (1) 

where c denotes the velocity of the stream in the direction of ^-positive, the 
method of Art. 33, when applied to a closed circuit, gives 

jy + fjL\f(uda; + vdy + wdz)=0, (2) 

whence j{udx + vdy + wdz) = Ge'^ (3) 

Hence the circulation in a circuit moving with the fluid, if once zero, is always 
zero. We now have 

- = const. - gy + /jl (ex + $) — J<? a , (4) 

this being, in fact, the form assumed by Art. 21 (2) when we write 

0, = gy-fi(cx-h(f>) (5) 

in accordance with (1) above. 

To calculate, in the first place, the effect of a simple-harmonic distribution 
of pressure we assume 

- = - x + &e k y sin Jcx, J^ = — y + ft e k y coskx (6) 

* The first steps of the following investigation are adapted from a paper by Rayleigh, "The 
Form of Standing Waves on the Surface of Eunning Water," Proc. Lond. Math. Soc. xv. 69 
(1883) [Papers, ii. 258], being simplified by the omission, for the present, of all reference to 
Capillarity. The definite integrals involved are treated, however, in a somewhat more general 
manner, and the discussion of the results necessarily follows a different course. 

The problem had been treated by Popoff, "Solution d'un probleme sur les ondes permanentes," 
Liouville (2), iii. 251 (1858); his analysis is correct, but regard is not had to the indeterminate 
character of the problem (in the absence of friction), and the results are consequently not pushed 
to a practical interpretation. 



400 Surface Waves [chap, ix 

The equation (4) becomes, on neglecting the square of &/3, 

■£- = . . . — gy + fie ky (kc 2 cos kx -f fie sin kx) (7) 

This gives for the variable part of the pressure at the upper surface (^r = 0) 

p = p/3 {(kc 2 — g) cos kx + yuc sin kx), (8) 

which is equal to the real part of 

p8 (kc 2 - g - ific) e ikx . 
If we equate the coefficient of e ikx to C, we may say that to the pressure 

po^Ce*** (9) 

corresponds the surface-form 

w-sr^ft*. (io) 

where we have written k for g/c 2 , so that Iitjk is the wave-length of the free 
waves which could maintain their position against the flow of the stream. 
We have also put A t / C== / U i> f° r shortness. 

Hence, taking the real parts, we find that the surface-pressure 

p = Ccoskx (11) 

produces the wave -form 

~ (k — k) cos kx — /«<! sin kx „ _ N 

m-*o* — w=WTk (12) 

This shews that if jjl be small the wave-crests will coincide in position 
with the maxima, and the troughs with the minima, of the applied pressure, 
when the wave-length is less than 2ir//c; whilst the reverse holds in the 
opposite case. This is in accordance with a general principle. If we impress 
on everything a velocity — c parallel to x, the result obtained by putting 
/i! = in (12) is seen to be a special case of Art. 168 (14). 

In the critical case of k = k, we have 

gpy = -—. sinkx, (13) 

Pi 

shewing that the excess of pressure is now on the slopes which face down the 

stream. This explains roughly how a system of progressive waves may be 

maintained against our assumed dissipative forces by a properly adjusted 

distribution of pressure over their slopes. 

243. The solution expressed by (12) may be generalized, in the first 
place by the addition of an arbitrary constant to x, and secondly by a sum- 
mation with respect to k. In this way we may construct the effect of any 
arbitrary distribution of pressure, say 

Po=A°>) (14) 

with the help of Fourier's Theorem (Art. 238 (4)). 



242-243] Surface Distribution of a Stream 401 

We will suppose, in the first instance, that f(os) vanishes for all but 
infinitely small values of a?, for which it becomes infinite in such a way that 

P f{x)dx = P\ (15) 

J -oo' 

this will give us the effect of an integral pressure P concentrated on an 

infinitely narrow band of the surface at the origin. Replacing C in (12) by 

Pjir . &k, and integrating with respect to k between the limits and oo , we 

obtain 

kP f°° (k — tc)coskx — /xisin&# 77 /nn -. 

goy = — . j~i * 2 dk (16) 

»™ 7T Jo {k-Kf + fX^ 

If we put £=k + im, where k, m are taken to be the rectangular co-ordinates of a variable 
point in a plane, the properties of the expression (16) are contained in those of the complex 
integral 



/ 



e ix C 

f=i* < 17 > 



It is known that the value of this integral, taken round the boundary of any area 
which does not include the singular point (f=c), is zero. In the present case we have 
c = k + ifii , where < and m are both positive. 

Let us first suppose that x is positive, and let us apply the above theorem to the region 
which is bounded externally by the line m = and by an infinite semicircle, described with 
the origin as centre on the side of this line for which m is positive, and internally by a 
small circle surrounding the point (*, m). The part of the integral due to the infinite 
semicircle obviously vanishes, and it is easily seen, putting £ — c = re id , that the part due to 
the small circle is 

if the direction of integration be chosen in accordance with the rule of Art. 32. We thus 
obtain 

[0 fiikx /"oo gikx 

i — ; ^- N dh + I -j — -. — . dk-27rie i ( K + i ^) x =0, 

J -oDJC-iK+lpi) JO k-(K+lm) 

which is equivalent to 

/"oo pikx /"oo fi-ikx 

Jo k-(K+im) Jo A: + (k + i/xi) 

On the other hand, when x is negative we may take the integral (17) round the contour 
made up of the line ra=0 and an infinite semicircle lying on the side for which m is 
negative. This gives the same result as before, with the omission of the term due to the 
singular point, which is now external to the contour. Thus, for x negative, 



/"oo pikx fee p -ikx 

r-j- — v—.dk=\ y-i —dk (19) 

Jo £-(«c+l/ii) JoH(k+^/Xi) V 



An alternative form of the last term in (18) may be obtained by integrating round the 
contour made up of the negative portion of the axis of k, and the positive portion of the 
axis of m, together with an infinite quadrant. We thus find 

/"O qVcx Too p-mx 

I i — i — v~- — sdk+ \ / . . idm=0, 

J -azk-(K+iiLi) Jo zm-(/c + i/x 1 ) 

which is equivalent to 



Too e - ilex /"oo 

Jo £ + (k-M>i) ' ~ J 



fll+U 



dm (20) 



402 Surface Waves [chap, ix 

This is for x positive. In the case of x negative, we must take as our contour the 
negative portions of the axes of k, ra, and an infinite quadrant. This leads to 



fco p — ikx fco pvnx 

j-^ r-,dk= — ~ rdm, (21) 



as the transformation of the second member of (19). 

In the foregoing argument /^ is positive. The corresponding results for the integral 

pixg 

, r . s dc (22) 

are not required for our immediate purpose, but it will be convenient to state them for 
future reference. For x positive, we find 

,ikx fco p — ikx fco p-mx 

dm; (23) 



/ 



fco oikx fco Q — ikx Too 

jofc-(K-im) ~Jo £+(k-«>i) ~Jo 



m+ni + iK 
whilst, for x negative, 

oikx fco p-ikx 



fco oikx fco Q-xkx 

I -. — -. ;— r dk = - 2irie* (« - <Mi) * + j - — — dk 

jo*-(k-»/*i) Jok + (K-ifj 



fco omx 

= - 2rrie i (*- i i*i) x + \ dm (24) 

Jo m — pi — iK 



The verification is left to the reader*. 

If we take the real parts of the formulae (18), (20), and (19), (21), respectively, we 
obtain the results which follow. 

The formula (16) is equivalent, for x positive, to 

trap rt „ . f°° (k + k) cos kx — iii sin kx 77 

kP * Jo (& + «) +A*1 

o ,. *• • , f 00 (m - vi) e~ mx dm /oex 

= -2ire-WsmKX+ ±- ™ , (25) 

Jo (m-fi^f+tc 2 

and, for x negative, to 

1T9P „_ l m (m-h H )e mx dm . 

KP' y -J (m + rip + * K0) 

The interpretation of these results is simple. The first term of (25) 
represents a train of simple-harmonic waves, on the down-stream side of the 
origin, of wave-length 2irc?lg, with amplitudes gradually diminishing according 
to the law e~^ x . The remaining part of the deformation of the free surface, 
expressed by the definite integrals in (25) and (26), though very great for 
small values of x, diminishes very rapidly as x increases in absolute value, 
however small the value of the- frictional coefficient fi lt 

When ixx is infinitesimal, our results take the simpler forms 
Trap _ . [ m coskx 



dk 



f °° Ytl6~ mx 

— — 2tt sin kx 4- — % ^dm t (27) 

JO Wl T * 



* For another treatment of these integrals, see Dirichlet, Vorlesungen ueber d. Lehre v. d. 
einfachen u. mehrfachen bestimmten Integralen (ed. Arendt), Braunschweig, 1904, p. 170. 



243-244] Surface Disturbance of a Stream 403 

for x positive, and 

cos Am? 



<irgp f 00 cos for 77 f°° me mx , ,_ ox 

kP u Jo k + /c Jo m a + « 2 v 7 



for x negative. The part of the disturbance of level which is represented 
by the definite integrals in these expressions is now symmetrical with respect 
to the origin, and diminishes constantly as the distance from the origin 
increases. When kx is moderately large we find, by usual methods, the 
semi-convergent expansion 

(•00 me -mx l 3 J 5 j 

^ s =-srr 2 --4i4+ -ins- ( 29 ) 



Jo m 2 +K 2 tc 2 x 2 k*x* k q x 6 

It appears bhat at a distance of about half a wave-length from the origin, on 
the down-stream side, the simple-harmonic wave-profile is fully established. 

The definite integrals in (27) and (28) can be reduced to known functions as follows. 
If we put (k + k) x=u, we have, for x positive, 

/°° cos kx _, _ Z" 00 cos {kx — u), 
o k+K ~ J KX u 

= -CiKXCOHicx + (%ir-Si <x)smKX, (30) 

where, in conformity with the usual notation, 

Ciw=-| du, Siu=l du (31) 

Ju u Jo u 

The functions Ci u and Si u have been tabulated by Glaisher*. It appears that as u 
increases from zero they tend very rapidly to their asymptotic values and £77, respectively. 



For small values of u we have 



Ci u=y + log u — 



2.2! " 4.4! 



SlU=*U — 



V? u 



.(32) 



3.3! ' 5.5! ' 

where y is Euler's constant *5772.... 

It is easily found from (25) and (26) that when fi\ is infinitesimal, the 
integral depression of the surface is 

-[" **»-£, (33) 

j -00 yp 

exactly as if the fluid were at rest. 

244. The expressions (25), (26) and (27), (28) alike make the elevation 
infinite at the origin, but this difficulty disappears when the pressure, which 
we have supposed concentrated on a mathematical line of the surface, is 
diffused over a band of finite breadth. 

* "Tables of the Numerical Values of the Sine-Integral, Cosine-Integral, and Exponential 
Integral," Phil. Trans. 1870; abridgments are given by Dale and by Jahnke and Emde. The 
expression of the last integral in (27) in terms of the sine- and cosine-integrals was obtained, in a 

different manner from the above, by Schlomilch, " Sur 1' integrate d^finie / ^ ^e~^, 

J d 2 + a? 

xxxiii. (1846); see also De Morgan, Differential and Integral Calculus, London, 1842, p. 654, 

and Dirichlet, Vorlesungen, p. 208. 



404 Surface Waves [chap, ix 

To calculate the effect of a distributed pressure 

J>o =/(*), (34) 

it is only necessary to write x — a for x in (27) and (28), to replace P by 
f(a) 8a, and to integrate the resulting value of y with respect to a between 
the proper limits. It follows from known principles of the Integral Calculus 
that if p be finite the integrals will be finite for all values of a?. 

In the case of a uniform pressure p , applied to the part of the surface 
extending from — oc to the origin, we easily find by integration of (25), for 
x> 0, 

. *p f 00 e- mx dm 
gpy = -2p cos K x + ^j^ m * + H a > ( 3o ) 

where yui has been put = 0. Again, if the pressure p be applied to the part 
of the surface extending from to + oo , we find, for x < 0, 



/cpo f °° em x dm 



f°° em x dm /Q ~, 

J. isrn? (36) 



From these results we can easily deduce the requisite formulae for the case 
of a uniform pressure acting on a band of finite breadth. The definite 
integral in (35) and (36) can be evaluated in terms of the functions Ci u, 
Si u ; thus in (35) 

[™e- mx dm r 00 sin foe,, rl a . N , n . . . /Qh .. 

/c — s 5 = i dk = (*7r — Si ##) cos /ca? + Ci /e# sin /c^c. . . .(37) 

Jo m 2 + « 2 Jo H« 

In this way the diagram on p. 405 was constructed ; it represents the case 
where the band (AB) has a breadth k~\ or 159 of the length of a standing 
wave. 

The circumstances in any such case might be realized approximately by 
dipping the edge of a slightly inclined board into the surface of a stream, 
except that the pressure on the wetted area of the board would not be uniform, 
but would diminish from the central parts towards the edges. To secure 
a uniform pressure, the board would have to be curved towards the edges, to 
the shape of the portion of the wave-profile included between the points 
A, B in the figure. 

It will be noticed that if the breadth of the band be an exact multiple 
of the wave-length (27r//c), we have zero elevation of the surface at a distance, 
on the down-stream as well as on the up-stream side of the source of 
disturbance. 

The diagram shews certain peculiarities at the points A, B due to the 
discontinuity in the applied pressure. A more natural representation of a 
local pressure is obtained if we assume 

p>=ivr* < 38 > 



244] 



Wave-Profile 



405 



406 Surface Waves [chap, ix 

We may write this in the form 

PI PC™ 
p = i-._!^ = £ e~ kb+ikx dk, (39) 

r 7T 6 — IX 7tJ 

provided it is understood that, in the end, only the real part -is to be retained. 
On reference to Art. 242 (9), (10), we see that the corresponding elevation of 
the free surface is given by 

„ p r<x> p— kb+ikx 

gpy = —\ .— r-dk (40) 

By the method of Art. 243, we find that this is equivalent, for x > 0, to 



„.p ( foo p—imb—mx 

gp y = ?£-\ 2irie^ +i ^ <**"*> 4- — dm\ , (41) 



and, for a? < 0, to 

0py = — — ■ —dm (42) 

Hence, taking real parts, and putting ^ = 0, we find 

r. r» * • «■? f °° w* cos mb — k sin mb m „ ■. r ~ n 

grpy = - 2fcPe- Kh sin k# H s s e~ mx dm, [x > 01, 

yrj/ 7r J o m 2 + tc 2 L J 

(43) 

/cP f 00 m cos mb — K sin m& „,„ 7 r „-. 

w=~J o ^^ *"*». [*«>]. 

(44) 

The factor e~ Kb in the first term of (43) shews the effect of diffusing the 

pressure. It is easily proved that the values of y and dy/dx given by these 

formulae agree when x—0*. 

245. If in the problem of Art. 242 we suppose the depth to be finite and 
equal to h, there will be, in the absence of dissipation, indeterminateness or 
not, according as the velocity c of the stream is less or greater than (gh)%, the 
maximum wave-velocity for the given depth; see Art. 229. The difficulty 
presented by the former case can be evaded by the introduction of small 
frictional forces; but it m&y be anticipated from the preceding investigation 
that the main effect of these will be to annul the elevation of the surface 
at a distance on the up-stream side of the region of disturbed pressure, 
and if we assume this at the outset we need not complicate our equations by 
retaining the frictional terms f. 

For the case of a simple-harmonic distribution of pressure we assume 
0. 



.(1) 



■#+j3 cosh k (y + h) sin hx, 

— = — ;/+/3 sinh k (y + h) cos kx, 

* A different treatment of the problem of Arts. 243, 244 is given in a paper by Kelvin, "Deep 
Water Ship-Waves," Proc. R. S. Edin. xxv. 562 (1905) [Papers, iv. 368]. 

t There is no difficulty in so modifying the investigation as to take the frictional forces into 
account, when these are very small. 



244-245] Stream of Finite Depth 407 

as in Art 233 (3). Hence, at the surface 

y=£ sinh M cos &r, (2) 

we have — = - gy - \ (q 2 - c 2 ) = 8 {Tec 2 cosh kh-gsinhkh) cos kx, (3) 

p 

so that to the imposed pressure 

p =Ccoskx (4) 

will correspond the surface-form 

__C sinh kh , ,_, 

p ' kc 2 cosh kh—g sinh kh ' 

As in Art. 242, the pressure is greatest over the troughs, and least over the crests, of the 
waves, or vice versd, according as the wave-length is greater or less than that corresponding 
to the velocity c, in accordance with general theory. 
The generalization of (5) by Fourier's method gives 



_ P /"°° sinh kh cos kx „ , ft , 

* irp J o kc 2 cosh kh - g sinh kh 



as the representation of the effect of a pressure of integral amount P applied to a narrow 
band of the surface at the origin. This may be written 

ttoc 2 [*> cos (xu/h) , mx 

-r ■ y= }« uoothu-ghi<? da (7) 

Now consider the complex integral 

9 ixClh 

dt, (8) 



/; 



^cothC-; 

where £=u + iv. The function under the integral sign has a singular point at (=+ico, 
according as x is positive or negative, and the remaining singular points are given by the 
roots of 

tanh£_ c 2 

~1 gk (9) 

Since (6) is an even function of x, it will be sufficient to take the case of x positive. 

Let us first suppose that c 2 > gh. The roots of (9) are then all pure imaginaries ; viz. 
they are of the form ± ij3, where j3 is a root of 

¥*£ ™ 

The smallest positive root of this lies between and ^tt, and the higher roots approximate 
with increasing closeness to the values (s + ^) it, where s is integral. We will denote these 
roots in order by /3 , j3j, j8 2 , .... Let us now take the integral (8) round the contour made 
up of the axis of u, an infinite semicircle on the positive side of this axis, and a series of 

small circles surrounding the singular points {=ip , ifii, $3 2 , The part due to the 

infinite semicircle obviously vanishes. Again, it is known that if a be a simple foot of 
f(£) = the value of the integral 



/ 



/«>* 



taken in the positive direction round a small circle enclosing the point £=a is equal to 

-f§ <») 

Now in the case of (8) we have 

/'(a) = cotha-a(coth2 a -l)=i|^(l-^) + a'j, (12) 

whence, putting a=i/3 8 , the expression (11) takes the form 

27rB 8 e-f*s*/\ (13) 



/; 



408 Surface Waves [chap, ix 

where *. " wn w\ (14) 

P* " c 2 \ C 2 ) 
The theorem in question then gives 

f0 fiixu'h /"co e ixu/h ^ 

J -ooWcothw-^A/c 2 Joucothu-gh/c 2 o 

If in the former integral we write — m for m, this becomes 

cos (xulh) 7 "C 00 D a v/ fc /t/>\ 

— ^ — L_^ ^ = 7r 2 n B s e-Ps x ' h (16) 

o « coth u— ghj c 2 ° 

The surface-form is then given by 

y=j?-K B > e ~ f ' ,xlh (17) 

It appears that the surface-elevation (which is symmetrical with respect to the origin) 
is insensible beyond a certain distance from the seat of disturbance. 

When, on the other hand, c 2 <gh, the equation (9) has a pair of real roots ( + a, say), the 
lowest roots (±/3 ) of (10) having now disappeared. The integral (7) is then indeterminate, 
owing to the function under the integral sign becoming infinite within the range of 
integration. One of its values, viz. the ' principal value,' in Cauchy's sense, can however 
be found by the same method as before, provided we exclude the points £= ±o from the 
contour by drawing semicircles of small radius e round them, on the side for which v is 
positive. The parts of the complex integral (8) due to these semicircles will be 

g±iax/h 

where/' (a) is given by (12); and their sum is therefore equal to 

27r^lsiny, (18) 

where A= (19) 

The equation corresponding to (16) now takes the form 

( fa-e [co ] cos l X u\h) , . . (XX ^oo B . . . 

{ + Y ttt — l -~T- 2 du = -7rAsm- r + 7rX B g e~^ h , (20) 

(Jo J a+e) u coth u-gh/c 2 hi 8 ' v ' 

so that, if we take the principal value of the integral in (7), the surface-form on the side 
of x positive is 

y=_^ 8in T + 42 S r^«" P '** (21) 

pc* h pc* • . l 

Hence at a distance from the origin the deformation of the surface consists of the 
simple-harmonic train of waves indicated by the first term, the wave-length 2ttA/o being 
that corresponding to a velocity of propagation c relative to still water. 

Since the function (7) is symmetrical with respect to the origin, the corresponding 
result for negative values of x is 

'-p^T + J 2 !"*** < 22 > 

The general solution of our indeterminate problem is completed by adding to (21) and 
(22) terms of the form 

Ccos™ + Dsin™ (23) 



245-246] Irregularities in the Bed of a Stream 409 

The practical solution, including the effect of infinitely small dissipative forces, is obtained 
by so adjusting these terms as to make the deformation of the surface insensible at 
a distance on the up-stream side. We thus get, finally, for positive values of x, 

y--^Bh? + — -S"*.*-^. (24) 

9 pc 2 h pc 2 l 

and, for negative values of x, 

y-^2^.A-» (25) 

For a different method of reducing the definite integral in this problem we must refer 
to the paper by Kelvin cited below. 

246. The same method can be employed to investigate the effect on a 
uniform stream of slight inequalities in the bed*. 

Thus, in the case of a simple-harmonic corrugation given by 

y = —h + y cos kx, (1) 

the origin being as usual in the undisturbed surface, we assume 

— = - x + (a cosh ky + {$ sinh ky) sin kx, 

-.(2) 
— = - y + (a sinh ky + (3 cosh ky) cos kx. 

The condition that (1) should be a stream-line is 

y= - a sinh kh + /3 cosh kh (3) 

The pressure-fonnula is 

- = const. -gy + kc 2 (o cosh ky + /3 sin ky) cos kx, (4) 

approximately, and therefore along the stream-line yjs = 

- = const. 4- (kc 2 a —gfi) cos kx, 

so that the condition for a free surface gives 

kc 2 a -gp = Q (5) 

The equations (3) and (5) determine a and 0. The profile of the free surface is given by 

'■ ffw< " a]FfeiB w '*' (6) 

If the velocity of the stream be less than that of waves in still water of uniform depth h, 
of the same length as the corrugations, as determined by Art. 229 (4), the denominator is 
negative, so that the undulations of the free surface are inverted relatively to those of the 
bed. In the opposite case, the undulations of the surface follow those of the bed, but with 
a different vertical scale. When c has precisely the value given by Art. 229 (4), the solution 
fails, as we should expect, through the vanishing of the denominator. To obtain an 
intelligible result in this case we should be compelled to take special account of dissipative 
forces. 

The above solution may be generalized, by Fourier's Theorem, so as to apply to the 
case where the inequalities of the bed follow any arbitrary law. Thus, if the profile of the 
bed be given by 

y=-h+f(x)=-h+~f a> dk[ CO /(£) cos A (*-£)<*£, (7) 

7Tj J -00 

* Sir W. Thomson, "On Stationary Waves in Flowing Water," Phil. Mag. (5) xxii. 353, 
445, 517 (1886), and xxiii. 52 (1887) [Papers, iv. 270]. The effect of an abrupt change of level in 
the bed is discussed by Wien, Hydrodynamik, p. 201. 



410 Surface Waves [chap, ix 

that of the free surface will be obtained by superposition of terms of the type (6) due to 
the various elements of the Fourier-integral; thus 



If 00 f 



00 /(flooB*(*-e dt 

oo cosh kh — g/kc 2 . sinh kh 



In the case of a single isolated inequality at the point of the bed vertically beneath 
the origin, this reduces to 

_ Q /• COS&37 „ 

7T J o cosh kh — g/kc 2 . sinh kh 



irh) o 



ucos(xu/h) j_ m 



w cosh u — gh/c 2 . sinh u 

where Q represents the area included by the profile of the inequality above the general 
level of the bed. For a depression Q will of course be negative. 

The discussion of the integral 



I fi M *"*f (10) 

J C cosh C-ghjc 2 . sinh £ v } 



can be conducted exactly as in Art. 245. The function to be integrated differs only by the 
factor £/(sinh £) ; the singular points therefore are the same as before, and we can at once 
write down the results. 

Thus when c 2 > gh we find, for the surface- form, 

y=$2*B 8 -^- e*M\ (11) 

9 h 8 sin ft, ' v ' 

the upper or the lower sign being taken according as x is positive or negative. 
When c 2 <gh, the 'practical' solution is, for x positive, 



y- 



fA*M% + f2"B*e-**l> 9 (12) 

A sinh a h h i sin/3 8 

and, for x negative, y=-^'£ co B 8 -J-?- e^ x ' h (13) 

rt l sin pig 

The symbols a, /3 S , A, B 8 have here exactly the same meanings as in Art. 245 * 

247. We may calculate, in a somewhat similar manner, the disturbance 
produced in the flow of a uniform stream by a submerged cylindrical obstacle 
whose radius b is small compared with the depth/ of its axisf. The cylinder 
is supposed placed horizontally athwart the stream. 

We write 



/ b\ 
4>~-cx\l +;i)+J6 (!) 



where c denotes as before the general velocity of the stream, and r denotes 
distance from the axis of the cylinder, viz. 

r = V(^ + (y+/) 2 ), (2) 

* A very interesting drawing of the wave-profile produced by an isolated inequality in the bed 
is given in Kelvin's paper, Phil. Mag. (5) xxii. 517 (1886) [Papers, iv. 295]. 

+ The investigation is taken from a paper "On some cases of Wave-Motion on Deep Water," 
Ann. di matematica (3), xxi. 237 (1913). I find that the problem had been suggested by Kelvin, 
Phil. Mag. (6) ix. 733 (1905) [Papers, iv. 369]. 



246-247] Waves due to a Submerged Cylinder 411 

the origin being in the undisturbed level of the surface, vertically above the 
axis. This makes d<f)/dr = for r = b, provided x De negligible in the neigh- 
bourhood of the cylinder. 

We assume 

X=\ a(k)e^y sin kccdk, (3) 

Jo 

where a (k) is a function of k, to be determined. For the equation of the free 
surface, assumed to be steady, we put 



V 



= 8 (k) cos kxdk (4) 

Jo 



The geometrical condition to be satisfied at the free surface is 

-£-?£. < 5 > 

wherein we may put y = 0. Since (1) is equivalent to 

r°° 

(}) = -cx-b 2 c e- k( v+f> sin kacdk+x> ( 6 ) 

Jo 

for positive values of y +/, this condition is satisfied if 

b 2 ce-W + a(k) = cj3(k) (7) 

Again, the variable part of the pressure at the free surface is given by 

f 00 dy 

— — 9V — i° 2 — b 2 c 2 e~ kf cos kxkdk + c =* 

= — 9V "" i ° 2 ~ & 2c2 e _A;/ cos kxkdk + c a (&) cos kxkdk, (8) 

Jo Jo 

where terms of the second order in the disturbance have been omitted. This 
expression will be independent of x provided 

g/3(k) + kb 2 c 2 e- k f-kca(k) = (9) 

Combined with (7), this gives 

«(*)-£^P<*-V ^W = fr"^ » ( 10 ) 

where K = g/c 2 , (11) 

as in Art. 242. Hence 

^70 f°° ke~ k f cos kxdk 
77 = 26 2 ^— - 

--d>^<^^ w 



412 Surface Waves [chap, ix 

The integral is indeterminate, but if x be positive its principal value is equal 
to the real part of the expression 

iire- K ^ iKX + i\ -. dm (13) 

Jo im—/c 

Adopting this we have 

2b 2 f 
7) — 2 ,/ 2 — 2ir K,b 2 e~ Kf sin kx 

- 2*6* r (K Si ° mf ~ T— z mf) ^ dm (14) 

Jo m 2 + /c 2 

For large values of x the second term is alone sensible. 

Since the value of tj in (12) is an even function of x we must have, for 
x negative, 

2 &Y , o z,2 - K f - o z.2 f°°(« sin rnf-m cos mf)e mx , v 

^= 2 ^ + %Tr/cb 2 e v sm k x- 2 Kb 2 \ v * — ^ , — — — am. ...(15) 

x 2 +f 2 Jo ra 2 +/e 2 

On the disturbances represented by these formulae we can superpose any 
system of stationary waves of length 2ir/fc, since these could maintain their 
position in space, in spite of the motion of the stream ; and if we choose as 
our additional system 

7] = — 2tt Kb 2 e~ K f sin kx (16) 

we shall annul the disturbance at a distance on the up-stream side (x < 0), as 
is required for a physical solution. The result is 

2b 2 f 
7] = -g — ~ - ^iTKb 2 e~ Kf sin kx -f &c. [x > 0], 

It appears that there is a local disturbance immediately above the obstacle, 
followed by a train of w$ves of length 2irc 2 lg on the down-stream side *. 

The investigation is easily adapted to the case where the section of the cylinder has any 
arbitrary form. The assumption really made above is that, to a first approximation, the 
effect of the cylinder at a distance is that of a suitably adjusted double source. In the more 
general case, referring to Art. 72 a, we may write 

4>= -ftp + ^ + x, (18) 

*--<%&$&&• < 19) 

It is convenient to work with complex quantities, and to write 



.(17) 



1 =cf C °e- l (i/+/) + «*rfi (20) 



with g= ^ + <?)-g (21) 

2tt 

* If we investigate the asymptotic expansion of the definite integral in (13), when k/ is large, 
we find on substitution in (12) that the most important term gives - 2b 2 //(:r 2 +/ 2 ), and so cancels 
the first term in the above values of r). The approximation has been carried further, for moderate 
values of k/, by Havelock, Proc. Roy. Soc. A, cxv. 274 (1927). 



247-248] Effect of a Travelling Disturbance 413 

The real part of (20) is of course alone to be retained in the end. The steps of the calcu- 
lation may be supplied by the reader. The final result is, for large values of | x |, 



■ ( A+ .f/ ^ X -{2(A + Q)KsmKX + 2KHcosKx}e-*f+&c. [a?>0], 



(22) 



The local disturbance near the origin is not symmetrical unless H=0. 

For an elliptic section whose major axis makes an a.ngle a with the direction of the stream, 
we have 

^l = 7 r(a 2 sin* 2 a + 6 2 cos 2 a), Q = nab, H= it (a 2 - 6 2 ) sin a cos a (23) 

The square of the amplitude of the waves is then 

4/c 2 (^ + ^) 2 + 4/c 2 .e' 2 = 47r 2 K 2 (a + 6) 2 (a 2 sin 2 a + 6 2 cos 2 a) (24) 

248. If in the problems of Arts. 243, 245 we impress on everything a 
velocity — c parallel to x, we get the case of a pressure-disturbance advancing 
with constant velocity c over the surface of otherwise still water. In this form 
of the question it is not difficult to understand, in a general way, the origin 
of the train of waves following the disturbance. 

If, for example, equal infinitesimal impulses be applied in succession to 
a series of infinitely close equidistant parallel lines of the surface, at equal 
intervals of time, each impulse will produce on its own account a system of 
waves of the character investigated in Art. 239. The systems due to the 
different impulses will be superposed, with the result that the only parts 
which reinforce one another will be those whose wave-velocity is equal to the 
velocity c with which the disturbing influence advances over the surface, and 
which are (moreover) travelling in the direction of this advance. And the 
investigations of Arts. 236, 237 shew that in the present problem the groups 
of waves of this particular length which are produced are continually being 
left behind. When capillary waves come to be considered, the latter statement 
will need to be modified. 

The question can be investigated from a general standpoint, independent 
of the particular kind of waves considered, as follows *. 

We take the origin at the instantaneous position of the disturbing influence, 
which is supposed to travel with velocity c in the direction of ^-negative. The 
effect of an impulse Bt delivered at an antecedent time t is given by Art. 241 (7) 
if we replace x by ct — x and multiply by St. Introducing $he hypothesis of a 
small frictional force varying as the velocity, and integrating from t = to 
t—QC, we get 



v =±- l w | r $ (k) e i7t ~ ik < c< -*> dk + j °°0 (k)e iTt+ik < c '-*> dk\ e^* dt. . . .(1) 

integration with respect to t gives 

_ 1 f 00 cj>(k)e ikx dk 1 p (f>(k)e~ ikx dk 

2irJo \ii — i{(T — kc) 27rJ \i*< — i(<r + kc) ' 



* Phil. Mag. (6) xxxi. 386 (1916). 



414 Surface Waves [chap, ix 

The quantity //- is by hypothesis small, and will in the limit be made to vanish. 

The most important part of the result will therefore be due to values of k in 

the first integral which make 

5 * = kc (3) 

approximately. Writing k = /c + k', where k is a root of this equation, we have 

a-kc = (^-cy , = {U-c)k f , (4) 

nearly, where U denotes the group- velocity corresponding to the wave-length 
2tt/k. The important part of (2) for large values of x is therefore 

1 f oo pik'x /JL' 

'-ir + W-L b-W-OV < 5 > 

since the extension of the range of integration to k' — ± oo makes no serious 
difference. Now if a be positive we have* 

roo e imx^ m ^ Mire-**, [x>0] 

J_ooa-Mm ( 0, |><0] {) 

whiist j^^fa"ishrt- [«<o] (7) 

Hence if U<c 

W)*™ e-triie-U) or o, (8) 

c— U 

according as x ^ ; whilst if U> c 

v = 0, or ^^e-W(^) (9) 

in the respective cases. If we now make yu. -^ we have the simple expression 

^ = -|7^Ff (10) 

for the wave-train generated by the travelling disturbance. This train follows 
or precedes the disturbing agent acccording as U $ c. Examples of the two 
cases are furnished by gravity waves on water, and capillary waves, respectively 
(Arts. 236, 266). 

The approximation in (4) is valid only if the quotient 

d*a/dk\k'~r-(U-c) (11) 

is small even when k'x is a moderate multiple of 27r. This requires that 

d*cr/dk* + (U-c)x (12) 

should be small. Unless JJ—c, exactly, the condition is always fulfilled if 
x be sufficiently great. It may be added that the results (8), (9) are accurate, 
in the sense that they give the leading term in the evaluation of (2) by 
Cauchy's method of residues. Cf. Art. 242. 

* The results quoted are equivalent to the familiar formulae 



/" cos mx dm _ f" 



m sin mx dm a^ 
1 tf-Htf =" 



(where the upper or lower sign is to be chosen according as x is positive or negative), but can be 
obtained directly by contour integration. 



248-249] Wave-Resistance 415 

In the case of waves on deep water, due to a concentrated pressure of 

integral amount P, we put 

<f>(k) = icrP/gp, (13) 

to conform to Art. 239 (28). Since U = Jc, we obtain, on taking the real part, 

?7 = sin kx, (14) 

9P 
in agreement with (27) of Art. 243*. 

If there is more than one value of k satisfying (3), there will be a term of 
the type (10) for each such value. This happens in the case of water-waves 
due to gravity and capillarity combined (Art. 269), and in the case of super- 
posed fluids, to be referred to presently. 

249. The preceding results have a bearing on the theory of ' wave-resistance.' 
Taking the two-dimensional form of the question, let us imagine two fixed 
vertical planes to be drawn, one in front, and the other in the rear, of the 
disturbing body. If U < c the region between the planes gains energy at the 
rate cE, where E is the mean energy per unit area of the free surface. This is 
due partly to the work done at the rear plane, at the rate UE (Art. 237), and 
partly to the reaction of the disturbing body. Hence if R be the resistance 
experienced by the latter, so far as it is due to the formation of waves, we have 

Rc+UE = cE, or R=°^E. (1) 

c 

On the other hand, if U > c, so that the wave-train precedes the body, the 
space between the planes loses energy at the rate cE. Since the loss at the 
first plane is UE, we have 

Rc-UE=-cE, or R=^^E. (2) 

c 

Thus, in the case of a disturbance advancing with velocity c [< y/(gh)] over 
still water of depth h, we find, on reference to Art. 237, 



R 



-M 1 -!™)' (3) 



where a is the amplitude of the waves. As c increases from to y/(gh), kK 
diminishes from oo to 0, so that R diminishes from %gpa 2 to 0. When 
c > *J(gh), the effect is merely local, and R = f. It must be remarked, 
however, that the amplitude a due to a disturbance of given type will also 
vary with c. For instance, in the case of the submerged cylinder, Art. 244 (43), 
a varies as ice~ Kh , where k = gjc 2 , the depth being infinite. Hence R varies as 

c -4g-2^/c2 ( 4 )J 

* It is not difficult to derive from (2) the complete formula referred to. 

f Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular 
Lectures and Addresses, London, 1889-94, iii. 450]. A formula equivalent to (3) was given in a 
paper by the same author, Phil. Mag. (5) xxii. 451 [Papers, iv. 279]. 

X The vertical force on the cylinder is calculated by Havelock, Proc. Roy. Soc. A, cxxii. 387 
(1928). 



416 



Surface Waves 



[chap. IX 



An interesting variation of the general question is presented when we have a layer of 
one fluid on the top of another of somewhat greater density. If p, p' be the densities of 
the lower and upper fluids, respectively, and if the depth of the upper layer be A', whilst 
that of the lower fluid is practically infinite, the results of Stokes quoted in Art. 231 shew 
that two wave-systems may be generated, whose lengths (27r//«) are related to the velocity c 
of the disturbance by the formulae 

C >=Z, <*= P-P' , i (5, 

K pCOth K/l+p K 

It is easily proved that the value of < determined by the second equation is real only if 



c*< 



p-p 



.(6) 



If c exceeds the critical value thus indicated, only one type of waves will be generated, 
and if the difference of densities be slight the resistance will be practically the same as in 
the case of a single fluid. But if c fall below the critical value, a second type of waves 
may be produced, in which the amplitude at the common boundary greatly exceeds that 
at the upper surface ; and it is to these waves that the ' dead-water resistance ' referred to 
in Art. 231 is attributed*. 

The problem of the submerged cylinder (Art. 247) furnishes an instance where the 
wave-resistance to the motion of a solid can be calculated. The mean energy, per unit 
area of the water surface, of the waves represented by the second term in equation (14) of 
that Art. is 

E=\gp(£TTK&e-*f)\ 
Since U=\c, we have from (1) 

R^AntgpbWe-W. (7) 

For a given depth (/) of immersion, this is greatest when <f— 1, or 

, , , . t c-J(gf) (8) 

In terms of the velocity c we have 

R=47rYpbK c-te-W* (9) 

The graph of R as a function of c is appended f. 
R 




*(gt) 



* Ekman, I.e. ante p. 371. 
t Ann. di mat.. I.e. 



See also the paper by the author, there quoted. 



249-250] Wave-Resistance 417 

Waves of Finite Amplitude. 

250. The restriction to ' infinitely small ' motions, in the investigations of 
Arts. 227, ..., implies that the ratio (ajX) of the maximum elevation to the 
wave-length must be small. The determination of the wave-forms which 
satisfy the conditions of uniform propagation without change of type, when 
this restriction is abandoned, forms the subject of a classical research by 
Stokes* and of many subsequent investigations. 

The problem is most conveniently treated as one of steady motion. It was 
pointed out by Rayleighf that if we neglect small quantities of the order 
a 3 /\ 3 , the solution in the case of infinite depth is contained in the formulae 

* == - x + ffe^ sin kx, X = - y + $#v cos Jew (1) 

The equation of the wave-profile (yfr = 0) is found by successive approxima- 
tions to be 

y = fie ky cos kx = (1 4- hy -f \k 2 y 2 + . . .) cos kx 
= P/3 2 + /3 (1 + |A^/3 2 ) cos kx + P/3 2 cos 2kx + p 2 /3 3 cos Skx + ... ; . . .(2) 

or, if we put /3 (1 + f & 2 /3 2 ) = a, 

y — \ ka 2 = a cos kx 4- 1 ka 2 cos 2&a> -f f& 2 a 3 cos Skx + ,(3) 

So far as we have developed it, this coincides with the equation of a trochoid, 
in which the circumference of the rolling circle is 2ir/k, or \ and the length 
of the arm of the tracing point is a. 

We have still to shew that the condition of uniform pressure along this 
stream-line can be satisfied by a suitably chosen value of c. We have, from 
(1), without approximation, 

2 = const. -gy-\c 2 {\- 2k/3e k v cos kx + k 2 /3 2 e 2k y}, (4) 

and therefore, at points of the line y — /3e k v cos kx, 

£ = const. + (kc 2 -g)y- \Wc 2 $ 2 e 2k y 

r 

= const. + (kc 2 — g — k 3 c 2 /3 2 ) y+ (5) 

Hence the condition for a free surface is satisfied, to the present order of 
approximation, provided 

? 2 = | + ^c 2 y8 2 = |(l+A; 2 a 2 ) (6) 



c< 



* "On the theory of Oscillatory Waves," Gamb. Trans, viii. (1847) [Papers, i. 197]. The 
method was one of successive approximation based on the exact equations of Arts. 9 and 20 ante. 
In a supplement of date 1880 the space-co-ordinates x, y are regarded as functions of the inde- 
pendent variables <p, \f/ [Papers, i. 314]. 

t I.e. ante p. 260. The method was subsequently extended so as to include all Stokes' results, 
Phil. Mag. (6) xxi. 183 [Papers, vi. 11]. 



41 8 Surface Waves [chap, ix 

This determines the velocity of progressive waves of permanent type, and 
shews that it increases somewhat with the amplitude a. 

The figure shews the wave-profile, as given by (3), in the case of ha = £, 
ora/\ = -0796* 



The approximately trochoidal form gives an outline which is sharper near 
the crests, and flatter in the troughs, than in the case of the simple-harmonic 
waves of infinitely small amplitude investigated in Art. 229, and these 
features become accentuated as the amplitude is increased. If the trochoidal 
form were exact, instead of merely approximate, the limiting form would 
have cusps at the crests, as in the case of Gerstner's waves to be considered 
presently. 

In the actual problem, which is one of irrotational motion, the extreme 
form has been shewn by Stokes f, in a very simple manner, to have sharp 
angles of 120°. The question being still treated as one of steady motion, 
the motion near the angle will be given by the formulae of Art. 63; viz. if 
we introduce polar co-ordinates r, 6 with the crest as origin, and the initial 
line of 6 drawn vertically downwards, we have 

ty = Cr m cos m6, (7) 

with the condition that \jr = when 0=±a (say), so that ma — \ir. This 
formula leads to 

Q = m Cr™~\ (8) 

where q is the resultant fluid-velocity. But since the velocity vanishes at the 
crest, its value at a neighbouring point of the free surface will be given by 

(f — 2gr cos a, (9) 

as in Art. 24 (2). Comparing (8) and (9), we see that we must have m = f, 
and therefore a = \tt\. 

In the case of progressive waves advancing over still water, the particles 
at the crests, when these have their extreme forms, are moving forwards with 
exactly the velocity of the wave. 

Another point of interest in connection with these waves of permanent 
type is that they possess, relatively to the undisturbed water, a certain 

* The approximation in (3) is hardly adequate for so large a value of ka ; see equation (17) 
below. The figure serves however to indicate the general form of the wave-profile. 

t Papers, i. 227 (1880). 

X The wave-profile has been investigated and traced by Michell, "The Highest Waves in 
Water," Phil. Mag. (5) xxxvi. 430 (1893). He finds that the extreme height is 142 \, and that 
the wave-velocity is greater than in the case of infinitely small height in the ratio of 1-2 to 1. See 
also Wilton, Phil. Mag. (6) xxvi. 1053 (1913). 



25o] Waves of Finite Height 419 

momentum in the direction of wave -propagation. The momentum, per wave- 
length, of the fluid contained between the free surface and a depth h (beneath 
the level of the origin), which we will suppose to be great compared with \, is 



-'/. 



dxdy = pch\, (10) 



since yjr = 0, by hypothesis, at the surface, and = ch, by (1), at the great depth 

h. In the absence of waves, the equation to the upper surface would be 

y = %ka 2 , by (3), and the corresponding value of the momentum would 

therefore be 

pc{h + \ka 2 )X (11) 

The difference of these results is equal to 

irpa 2 c, (12) 

which gives therefore the momentum, per wave-length, of a system of 
progressive waves of permanent type, moving over water which is at rest at 
a great depth. 

To find the vertical distribution of this momentum, we remark that the 
equation of a stream-line ^ — ch' is found from (2) by writing y-\-ti for y 
and {3e~ kh ' for (3. The mean-level of this stream-line is therefore given by 

y=-h' + ik/3 2 e- 2kh ' (13) 

Hence the momentum, in the case of undisturbed flow, of the stratum of 
fluid included between the surface and the stream-line in question would 
be, per wave-length, 

pc\{h' + P/S a (l-e- M *')} (14) 

The actual momentum being pch'X, we have, for the momentum of the same 
stratum in the case of waves advancing over still water, 

7rpft 2 c(l-e- 2M ') (15) 

It appears therefore that the motion of the individual particles, in these 
progressive waves of permanent type, is not purely oscillatory, and that there 
is, on the whole, a slow but continued advance in the direction of wave- 
propagation*. The rate of this flow at a depth K is found approximately by 
differentiating (15) with respect to h! ', and dividing by p\, viz. it is 

WaPoer**' (16) 

This diminishes rapidly from the surface downwards. 

The further approximation by Stokes, confirmed by the independent cal- 
culations of Rayleigh and others, gives as the equation of the wave-profile 

y = const. + a cos hx — {\ka 2 + JjA^a 4 ) cos 2kx + f Pa 3 cos Skx 

- iPa 4 cos4&#+..., (17) 

* Stokes, I.e. ante p. 417. Another very simple proof of this statement has been given by 
Rayleigh, I.e. ante p. 260. 



420 Surface Waves [chap, ix 

with, for the wave-velocity, 

c 2 = |(l + A; 2 a 2 + J& 4 a 4 +...) (18) 

A question as to the convergence, both of the series which form the coeffi- 
cients of the successive cosines when the approximation is continued, and of 
the resulting series of cosines, was raised by Burnside*, who even expressed 
a doubt as to the possibility of waves of rigorously permanent type. This led 
Rayleigh to undertake an extended investigation f, which shewed that the 
condition of uniformity of pressure at the surface could be satisfied, for 
sufficiently small values of ha, to a very high degree of accuracy. He inferred 
that the existence of permanent types up to the highest wave of Michell was 
practically, if not demonstrably, certain. The existence has at length been 
definitely established by an investigation of Prof. Levi CivitaJ, which puts an 
end to an historic controversy. 

There are one or two simple properties of these permanent waves which come easily 
from first principles § . The problem being reduced to one of steady motion, let the origin 
be taken in the mean level, beneath (say) a crest, and let X be the wave-length. Denoting 
by rj surface-elevation above the mean level, we have, then, 

ndx=0 (19) 



/. 



Also, if q be the surface velocity, and q its value at the mean level, we have 

q 2 = q 2 -2grj, 

and therefore / q i dx = q<?\ (20) 

Jo 

Again, consider the mass of fluid contained between vertical planes through two successive 
crests, and bounded below by a plane y= —h x at which the velocity is sensibly horizontal 
and equal to c. It is easily seen that the total vertical mass-acceleration is zero, since there 
is no flux of vertical momentum across the boundaries. Hence if p be the surface -pressure, 
and p x that at the depth h x , 



/ (Pi-P)dx=gpj (h 1 + r } )dx=gph 1 \ (21) 



But, comparing pressures in the same vertical we have 

Pi-p=9p(h\+ii)+\{q 2 -<?\ 

f x 

and thence / q 2 dx=c 2 X (22) 

Jo 

We may express this by saying that the mean square of the surface velocity, per equal 
increments of #, is equal to c 2 . It follows also from (20) that q = c, i.e. the velocity at the 
points where the wave-profile meets the mean level is equal to c. 

* Proc. Lond. Math. Soc. (2) xv. 26 (1916). 

t Phil. Mag. (6) xxxiii. 381 (1917) [Papers, vi. 478]. 

$ "Determination rigoureuse des ondes permanentes d'ampleur finie," Math. Ann. xciii. 264 
(1925). The extension to waves in a canal of finite depth has been made by Struik, Math. Ann. xcv. 
595 (1926). 

§ Levi Civita, I.e. 



25(>-25i] Gerstner's Rotational Waves 421 

251. A system of exact equations, expressing a possible form of wave- 
motion when the depth of the fluid is infinite, was given so long ago as 1802 
by Gerstner*, and at a later period independently by Rankinef. The circum- 
stance, however, that the motion in these waves is not irrotational detracts 
somewhat from the physical interest of the results. 

If the axis of x be horizontal, and that of y be drawn vertically upwards, 
the formulae in question may be written 

x = a + y e*** sin k (a + ct), y = 6— ye kb cos k (a + ct), (1) 

where the specification is on the Lagrangian plan (Art. 16), viz. a, b are two 
parameters serving to identify a particle, and x, y are the co-ordinates of this 
particle at time t The constant k determines the wave-length, and c is the 
velocity of the waves which are travelling in the direction of ^-negative. 

To verify this solution, and to determine the value of c, we remark, in the 
first place, that 

d(a,b) 6 ' W 

so that the Lagrangian equation of continuity (Art. 16 (2)) is satisfied. Again, 
substituting from (1) in the equations of motion (Art. 13), we find 

jr (- + 9y) = kc 2 e kb sin k (a + ct\ 

gi (- + 9V) = ~ kc 2 e kb cos k (a + ct) + kc 2 e m ; 
whence 

^ = const. - g \b - t e™> cos k (a + ct) I - c 2 e kb cos k(a + ct) + % c 2 e 2kb . . . .(4) 

For a particle on the free surface the pressure must be constant; this requires 

c 2 = g/k, (5) 

as in Art. 229. This makes 

V - = const. -#& + ic 2 e 2to (6) 

It is obvious from (1) that the path of any particle (a, b) is a circle ot 
radius k' 1 ^. 

It has already been stated that the motion of the fluid in these waves is 
rotational. To prove this we remark that 

= |S{e fc6 sinA;(a + cO} + ce 2fc& 8a, (7~ 

which is not an exact differential. 

* Professor of Mathematics at Prague, 1789-1823. His paper, "Theorie der Wellen," was 
published in the Abh. d. k. bohm. Ges. d. Wiss. 1802 [Gilbert's Annalen d. Physik, xxxii. (1809)]. 

t "On the Exact Form of Waves near the Surface of Deep Water," Phil. Trans. 1863 
[Papers, p. 481], 



•(3) 



422 Surface Waves [chap, ix 

The circulation in the boundary of the parallelogram whose vertices 
coincide with the particles 

(a, b), (a + 8a, b), (a, b + 8b), (a + 8a,b + 8b) 

is, therefore, - ^ {ce 2kb 8a) 8b, 

and the area of the circuit is 



l&^8a8b - (1 - e™) 8a8b. 
d (a, b) 



d 

Hence the vorticity (co) of the element (a, b) is 

2kce 2kb 
W = -]T^ < 8 ) 

This is greatest at the surface, and diminishes rapidly with increasing depth. 
Its sense is opposite to that of the revolution of the particles in their circular 
orbits. 

A system of waves of the present type cannot therefore be originated 
from rest, or destroyed, by the action of forces of the kind contemplated in 
the general theorem of Arts. 17, 33. We may however suppose that by 
properly adjusted pressures applied to the surface of the waves the liquid is 
gradually reduced to a state of flow in horizontal lines, in which the velocity 
(v!) is a function of the ordinate (y f ) only*. In this state we shall have 
doc' /da = 1, while y' is a function of b determined by the condition 

<j Wj_ y') d (x, y) . 

9 (a, b) d(a,b)' W 

or l^i-gto (10) 

This makes % =%%- - 2a> t' = 2te**», (11) 

cb dy' db db ' 

and therefore u = ce 2kb (12) 

Hence, for the genesis of the waves by ordinary forces, we require as a 

foundation an initial horizontal motion, in the direction opposite to that of 

propagation of the waves ultimately set up, which diminishes rapidly from 

the surface downwards, according to the law (12), where b is a function of y' 

determined by 

y'^b-lk- 1 ** (13) 

It is to be noted that these rotational waves, when established, have zero 
momentum. 

The figure shews the forms of the lines of equal pressure 6 = const., for 
a series of equidistant values of 6f. These curves are trochoids, obtained by 

* For a fuller statement of the argument see Stokes' Papers, i. 222. 

t The diagram is very similar to the one given originally by Gerstner, and copied more or less 
closely by subsequent writers. A version of Gerstner's investigation, including in one respect a 
correction, was given in the second edition of this work, Art. 233. 



251-252] 



Gerstner's Waves 



423 



rolling circles of radii Ar 1 on the under sides of the lines y=b + kr\ the 
distances of the tracing points from the respective centres being k" 1 ^. Any 
one of these lines may be taken as representing the free surface, the extreme 
admissible form being that of the cycloid. The dotted lines represent the 
successive forms taken by a line of particles which is vertical when it passes 
through a crest or a trough. 




252. Scott Russell, in his interesting experimental investigations*, was 
led to pay great attention to a particular type which he called the 'solitary 
wave.' This is a wave consisting of a single elevation, of height not necessarily 
small compared with the depth of the fluid, which, if properly started, may 
travel for a considerable distance along a uniform canal, with little or no 
change of type. Waves of depression, of similar relative amplitude, were 
found not to possess the same character of permanence, but to break up into 
series of shorter waves. 

Russell's 'solitary' type may be regarded as an extreme case of Stokes' 
oscillatory waves of permanent type, the wave-length being great compared 
with the depth of the canal, so that the widely separated elevations are 
practically independent of one another. The methods of approximation 
employed by Stokes become, however, unsuitable when the wave-length 
much exceeds the depth ; and subsequent investigations of solitary waves 
of permanent type have proceeded on different lines. 

The first of these was given independently by Boussinesqt and RayleighJ. The latter 
writer, treating the problem as one of steady motion, starts virtually from the formula 

d 

(1) 



iy 



<\> + ity = F(x + iy) = e dx F(x), 



* "Beport on Waves," Brit. Ass. Rep. 1844. 

t Comvtes Rendus, June 19, 1871. J I.e. ante p. 260. 



424 Surface Waves [chap, ix 

where F(x) is real. This is especially appropriate to cases, such as the present, where 
one of the family of stream-lines is straight. We derive from (1) 

0-y-f^+f^ 1 *-..., ♦-y^'-f^+fl^-..., (2) 

where the accents denote differentiations with respect to x. The stream-line \^ = here 
forms the bed of the canal, whilst at the free surface we have \//-= -cA, where c is the 
uniform velocity, and h the depth, in the parts of the fluid at a distance from the wave, 
whether in front or behind. 

The condition of uniform pressure along the free surface gives 

u 2 + v 2 =c 2 -2g(y-h), (3) 

or, substituting from (2), 

F' 2 -y 2 F'F"+y 2 F" 2 +... = c 2 -2g(y-h) (4) 

But, from (2) we have, along the same surface, 

yF'-f l F'"+...= -ck (5) 

It remains to eliminate F between (4) and (5) ; the result will be a differential equation 
to determine the ordinate y of the free surface. If (as we will suppose) the function F' (x) 
and its differential coefficients vary so slowly with x that they change only by a small 
fraction of their values when x increases by an amount comparable with the depth h, the 
terms in (4) and (5) will be of gradually diminishing magnitude, and the elimination in 
question can be carried out by a process of successive approximation. 
Thus, from (5), 

^— 7+s^+"— «»jj+«* , G)" + -} ! (6) 

and if we retain only terms up to the order last written, the equation (4) becomes 

y 2 3 y \y) ^ \y) h 2 c 2 h 2 ' 

or, on reduction, 

1 2y" \y' 2 _ 1 Zg(y-h) 

y 2 3 y 3y 2 h 2 c 2 h 2 {n 

If we multiply by y', and integrate, determining the arbitrary constant so as to make 
y = for y = k, we obtain 

1 \y' 2 _ 1 y-h g(y-h) 2 
y 3 y h^ h 2 c 2 h 2 ' 



or 



^fc^-f)- 



.(8) 



Hence y' vanishes only for y = h and y=c 2 /g, and since the last factor must be positive, 
it appears that c 2 \g is a maximum value of y. Hence the wave is necessarily one of eleva- 
tion only, and, denoting by a the maximum height above the undisturbed level, we have 

c 2 =g{h + a), (9) 

which is exactly the empirical formula for the wave-velocity adopted by Russell. 

The extreme form of the wave must, as in Art. 250, have a sharp crest of 120° ; and 
since the fluid is there at rest we shall have c 2 = 2ga. If the formula (9) were applicable to 
such an extreme case, it would follow that a = h. 

If we put, for shortness, 

h 2 (h + a) 19 

we find, from (8), 

'-±i0-9*. -<"> 



252] 



Solitary Wave 



425 



a sech 2 



.(12) 



the integral of which is 

if the origin of x be taken beneath the summit. 

There is no definite ' length ' of the wave, but we may note, as a rough indication of its 
extent, that the elevation has one-tenth of its maximum value when 0/6 = 3*636. 

\y 




x' O x 

The annexed drawing of the curve 

y = l+^sechH# 
represents the wave-profile in the case a=\h. For lower waves the scale of y must be 
contracted, and that of x enlarged, as indicated by the annexed table giving the ratio bjh, 
which determines the horizontal scale, for various values of a/A. 

It will be found, on reviewing the above investigation, that the approximations consist 
in neglecting the fourth power of the ratio (A + a)/2fe*. 

If we impress on the fluid a velocity — c parallel to x we get the case 
of a progressive wave on still water. It is not difficult to shew that, if the 
ratio alh be small, the path of each particle is then an arc of a parabola 
having its axis vertical and apex upwards f. 

It might appear, at first sight, that the above theory is inconsistent 
with the results of Art. 187, where it was argued that a wave of finite 
height whose length is great compared with the depth must inevitably 
suffer a continual change of form as it advances, the changes being the 
more rapid the greater the elevation above the undisturbed level. The 
investigation referred to postulates, however, a length so great that the 
vertical acceleration may be neglected, with the result that the horizontal 
velocity is sensibly uniform from top to bottom (Art. 169). The numerical table above 
given shews, on the other hand, that the longer the ' solitary wave ' is, the lower it is. In 
other words, the more nearly it approaches to the character of a ' long ' wave, in the sense 
of Art. 169, the more easily is the change of type averted by a slight adjustment of the 
particle- velocities % . 

The motion at the outskirts of the solitary wave can be represented by a very simple 
formula. Considering a progressive wave travelling in the direction of ^-positive, and 
taking the origin in the bottom of the canal, at a point in the front part of the wave, we 
assume 

(p = Ae~ m ^- ct ) cos my (13) 

This satisfies V 2 <£ = 0, and the surface-condition 

d<j) 



a/h 


bjh 


•1 


1-915 


•2 


1-414 


•3 


1-202 


•4 


1-080 


•5 


1-000 


•6 


•943 


•7 


•900 


•8 


•866 


•9 


•839 


1-0 


•816 



+9zz = ° 



dt* ' » dy 



.(14) 



* The theory of the solitary wave has been treated by Weinstein, Lincei (6) iii. 463 (1926), by 
the method of Levi Civita referred to in Art. 250. He finds that the formula (9) is a very close 
approximation. 

f Boussinesq, I.e. 

% Stokes^ "On the Highest Wave of Uniform Propagation," Proc. Camb. Phil. Soc. iv. 361 
(1883) [Papers, v. 140]. 



426 Surface Waves [chap, ix 

will also be satisfied for y= h, provided 

„ , tan mh 

° 2 =^ "1ST (15) 

This will be found to agree approximately with Kayleigh's investigation if we put m=b~ 1 . 

The above remark, which was communicated to the author by the late Sir George 
Stokes*, was suggested by an investigation by McCowant, who shewed that the formula 

izlz^ ^(x+iy)+ata,nh \m(x+iy) (16) 

satisfies the conditions very approximately, provided 

c 2 =- tanraA, (17) 

m 

and «ia=§sin 2 m (A + §a), a=atan^ra(A + a), (18) 

where a denotes the maximum elevation above the mean level, and a is a subsidiary 
constant. In a subsequent paper J the extreme form of the wave when the crest has a 
sharp angle of 120° was examined. The limiting value of the ratio a/h was found to be '78, 
in which case the wave-velocity is given by c 2 = 1 'bQgh. 

253. By a slight modification the investigation of Rayleigh and Boussinesq 
can be made to give the theory of a system of oscillatory waves of finite height 
in a canal of limited depth §. 

In the steady-motion form of the problem the momentum per wave-length (X) is repre- 
sented by 



llpudxdy = -pi I ^dxdy = -p^X, 



•(19) 



where ty x corresponds to the free surface. If h be the mean depth, this momentum may be 
equated to pch\, where c denotes (in a sense) the mean velocity of the stream. On this 
understanding we have, at the surface, y\r x = —ch, as before. The arbitrary constant in (3), 
on the other hand, must be left for the moment undetermined, so that we write 

u 2 + v 2 = C-2gy (20) 

We then find, in place of (8), 

y"=^y-i)^-y)(y-h2\ (21) 

where h x , h 2 are the upper and lower limits of y, and 

e 2 k 2 
l = ~ (22) 

It is implied that I cannot be greater than h 2 . 

If we now write y = ^i cos 2 ^ + A 2 sm2 X > (23) 

we find /3^ = V{l-£ 2 sin 2 x}, (24) 



* Cf. Papers, v. 62. 

t " On the Solitary Wave," Phil. Mag. (5) xxxii. 45 (1891). 

{ "On the Highest Wave of Permanent Type," Phil. Mag. (5) xxxviii. 351 (1894). 

§ Korteweg and De Vries, "On the Change of Form of Long Waves advancing in a Beet- 
angular Canal, and on a New Type of Long Stationary Waves," Phil. Mag. (5) xxxix. 422 (1895). 
The method adopted by these writers is somewhat different. Moreover, as the title indicates, the 
paper includes an examination of the manner in which the wave-profile is changing at any instant, 
if the conditions for permanency of type are not satisfied. 

For other modifications of Rayleigh 's method reference may be made to Gwyther, Phil. Mag. 
(5)1. 213, 308, 349 (1900). 



252-254] Solitary Wave 427 

'V{fi3>}; k2=h -& (26) 



Hence, if the origin of x be taken at a crest, we have 



Jo J 



<?x 



J(l -k 2 sin 2 x 
The wave-length is given by 



-/a^(x.*). ( 26 ) 



and y=A 2 +(^i-^2)cn 2 |. [mod. &] (27)* 



^W wa-^W 2 ^' - (28) 

Again, from (23) and (24), 

^-^ ^S^ ^-yWW+ft-OJkW) (29) 

Since this must be equal to AX, we have 

(h-l)F 1 (k) = (h 1 -l)E 1 (k) (30) 

In equations (25), (28), (30) we have four relations connecting the six quantities A 1} h 2i 
I, k, X, /3, so that if two of these be assigned the rest are analytically determinate. The 
wave-velocity c is then given by (22) t. For example, the form of the waves, and their 
velocity, are determined by the length X, and the height h x of the crests above the bottom. 

The solitary wave of Art. 252 is included as a particular case. If we put l=k 2 , we 
have &=1, and the formulae (28) and (30) then shew that X= oo , h 2 = h. 

254. The theory of waves of permanent type has been brought into rela- 
tion with general dynamical principles by Helmholtzj. 

If in the equations of motion of a 'gyrostatic' system, Art. 141 (23), we 

put 

dV dV dV 

*--£ *"£' •- Q "~£' (1) 

where V is the potential energy, it appears that the conditions for steady 
motion, with q 1% q 2> ... q n constant, are 

4(F + Z)-0, ^(V + K) = 0, .... £ n (V+mO, ...(2) 

where K is the energy of the motion corresponding to any given values of 
the co-ordinates q lt q 2i ... q n when these are prevented from varying by the 
application of suitable extraneous forces. 

This energy is here supposed expressed in terms of the constant momenta 
corresponding to the ignored co-ordinates %, %', ..., and of the palpable 
co-ordinates q lf q%, ..- q n - It may however also be expressed in terms of the 

* The waves represented by (27) are called 'cnoidal waves' by the authors cited. For the 
method of proceeding to a higher approximation we must refer to the original paper. 

f When the depth is finite, a question arises as to what is meant exactly by the • velocity of 
propagation.' The velocity adopted in the text is that of the wave-profile relative to the centre of 
inertia of the mass of fluid included between two vertical planes at a distance apart equal to the 
wave-length. Cf. Stokes, Papers, i. 202. 

J "Die Energie der Wogen und des Windes," Berl. Monatsber, July 17, 1890 [Wiss^ Abh. 
iii. 333] . 



428 Surface Waves [chap, ix 

velocities %, ^', ... and the co-ordinates q lt q 2 , ... q n \ in this form we denote 
it by Tq. It may be shewn, exactly as in Art. 142, that dT /dq r = — dK/dq r , so 
that the conditions (2) are equivalent to 

i {V - T ° )=0 - i (F - r ° )=0 ' -' 4^-^=0. ..,(3) 

Hence the condition for free steady motion with any assigned constant 
values of q lt q%, ... q n is that the corresponding value of V + K, or of V — T , 
should be stationary. Cf. Art. 203 (7). 

Further, if in the equations of Art. 141 we write — dV/dq r + Q r for Q r , so 
that Q r now denotes a component of extraneous force, we find, on multiplying 
by <7i, q 2 , ... q n in order, and adding, 

^.(®+ V + K)=Q 1 q 1 +Q 2 q 2 +...+Q n q ni (4) 

where ® is the part of the energy which involves the velocities q 1} q 2 , • •• qu- 
it follows, by the same argument as in Art. 205, that the condition for 
'secular' stability, when there are dissipative forces affecting the co-ordinates 
qi, q 2 , ... q n , but not the ignored co-ordinates X> X> -•-> * s tna ^ T^+if should 
be a minimum. 

In the application to the problem of stationary waves, it will tend to clearness if we 
eliminate all infinities from the question by imagining that the fluid circulates in a ring- 
shaped canal of uniform rectangular section (the sides being horizontal and vertical), of 
very large radius. The generalized velocity x corresponding to the ignored co-ordinate 
may be taken to be the flux per unit breadth of the channel, and the constant momentum 
of the circulation may be replaced by the cyclic constant k. The co-ordinates q u q 2 , ... q n 
of the general theory are now represented by the value of the surface-elevation (i/) 
considered as a function of the longitudinal space-co-ordinate x. The corresponding 
components of extraneous force are represented by arbitrary pressures applied to the 
surface. 

If I denote the whole length of the circuit, then considering unit breadth of the canal 
we have 

V=\gp^dx, (5) 

where rj is subject to the condition 

f rjdx^O (6) 

If we could with the same ease obtain a general expression for the kinetic energy of 
the steady motion corresponding to any prescribed form of the surface, the condition in 
either of the forms above given would, by the usual processes of the Calculus of Varia- 
tions, lead to a determination of the possible forms, if any, of stationary waves*. 

For some general considerations bearing on the problem of stationary waves on the common 
surface of two currents reference may be made to Helmholtz' paper. This also contains, at the 
end, some speculations, based on calculations of energy and momentum, as to the length of the 
waves which would be excited in the first instance by a wind of given velocity. These appear to 
involve the assumption that the waves will necessarily be of permanent type, since it is only on 
some such hypothesis that we get a determinate value for the momentum of a train of waves of 
small amplitude. 



254-255] Dynamical Condition for Permanent Type 429 

Practically, this is not feasible, except by methods of successive approximation, but 
we may illustrate the question by reproducing, on the basis of the present theory, the 
results already obtained for 'long' waves of infinitely small amplitude. 

If h be the depth of the canal, the velocity in any section when the surface is maintained 
at rest, with arbitrary elevation 77, is x/(k + 17), where % is the flux. Hence, for the cyclic 
constant, 

K =*j o v,)-^4*(i+^ o w), ( 7) 

approximately, where the term of the first order in 77 has been omitted, in virtue of (6). 

The kinetic energy, ^p<jc> may be expressed in terms of either % or k. We thus obtain 
the forms 

I ^* e y( 1+ mj>*)' < 8 > 

K -i P -¥( l -mfo" 2dx ) (9) 

The variable part of P— T is 

ip (g- pj | o V<&, (10) 

and that of 7+ K is 

*('-£)&*' (n) 

It is obvious that these are both stationary for 17 — ; and that they will be stationary 
for any infinitely small values of 77, provided ^ 2 =^A 3 , or < 2 =ghl 2 . If we put x = c ^i or 
k = cI, this condition gives 

c 2 =gh, (12) 

in agreement with Art. 175. 

It appears, moreover, that rj = makes V+ K a maximum or a minimum according as 
c 2 is greater or less than gh. In other words, the plane form of the surface is secularly 
stable if, and only if, c < *J(gh). It is to be remarked, however, that the dissipative forces 
here contemplated are of a special character, viz. they affect the vertical motion of the 
surface, but not (directly) the flow of the liquid. It is otherwise evident from Art. 175 
that if pressures be applied to maintain any given constant form of the surface, then if 
c 2 > gh these pressures must be greatest over the elevations and least over the depressions. 
Hence if the pressures be removed, the inequalities of the surface will tend to increase. 

Wave-Propagation in Two Dimensions. 

255. We may next consider some cases of wave-propagation in two 
horizontal dimensions x, y. The axis of z being drawn vertically upwards, we 
have, on the hypothesis of infinitely small motions, 

p _ d(p 
P 
where cp satisfies V 2 <£ = (2) 

The arbitrary function F(t) may be supposed merged in the value ofd<f>/dt 

If the origin be taken in the undisturbed surface, and if f denote the 
elevation at time t above this level, the pressure-condition to be satisfied at 
the surface is 

'-ffl~ <* 



--gz + F(t), (1) 



430 Surface Waves [chap, ix 

and the kinematical surface-condition is 

dt [>_Uo' w 



cf. Art. 227. Hence, for z = 0, we must have 



-si+tfsE-0, (5) 



a* 2 r *a* 

or, in the case of simple-harmonic motion, 

"> = ^ (6) 

if the time-factor be e i(Tt+e) . 

The fluid being supposed to extend to infinity, horizontally and down- 
wards, we may briefly examine, in the first place, the effect of a local initial 
disturbance of the surface, in the case of symmetry about the origin. 

The typical solution for the case of initial rest is easily seen, on reference 
to Art. 100, to be 

* = /^Wo(M,j (7) 

%= cos <rt Jo (kvr), J 
provided cr 2 = gk y (8) 

as in Art. 228. 

To generalize this, subject to the condition of symmetry, we have recourse 
to the theorem 

/(«)= rj (kvr)kdk r/(a)J (ka)ada (9) 

Jo Jo 

of Art. 100 (12). Thus, corresponding to the initial conditions, 

r=/(o, tf» =o, (io) 



.(ii) 



r<x> gij^ Q.+ roo \ 

we have ^—g] e kz Jo(k^)kdk\ f(a)J (ka)ada,) 

J o <r Jo I 

f— cos at Jo (kvr)kdk I f(a)J (ka)ada. 

If the initial elevation be concentrated in the immediate neighbourhood 
of the origin, then, assuming 

rf(a)2>rrada = l, (12) 

Jo 

we have 6 = ^-1 e kz J {k^)kdk ^l3) 

Ait J a 

Expanding, and making use of (8), we get 



255] Propagation in two Dimensions 431 

Ifweput z — -rcos6, «r = rsin0, (15) 

/•oo \ 

vfe have e^J (krff)dk = - , (16) 

Jo r 

by Art. 102 (9), and thence* 

^Ji(far)*-dfc-g)*J-»!^ f (17) 

where yu. = cos (of. Art. 85). Hence 

* = 2^ 1^^ 3T 73 + -5l J*—-- J" - (18) 

From this the value of f is to be obtained by (3). It appears from 
Arts. 84, 85 that 

p 2 „ + i(o) = o, fhW-t-)' 1 !"''^ 11 , (19) 

whence 

1 fi»^ V.&fgW. l'.ffl.ffl/gA' { , MV i 
?- 2^l2!^ irUJ + 10! W "-}■ - (2 ° )T 

It follows that any particular phase of the motion is associated with 
a particular value of gfi/tj, and thence that the various phases travel radially 
outwards from the origin, each with a constant acceleration. 

No exact equivalent for (20), analogous to the formula (21) of Art. 238 
which was obtained in the two-dimensional form of the problem, and accord- 
ingly suitable for discussion in the case where gt 2 /^ is large, has been dis- 
covered. An approximate value may however be obtained by Kelvin's method 
(Art. 241). Since Jo {z) is a fluctuating function which tends as z increases 
to have the same period 27r as sin z, the elements of the integral in (13) will 
for the most part cancel one another with the exception of those for which 

tdcr\dk=Ts, or km = gt 2 l^i^, (21) 

nearly. Now when km is large we have 

Mk&)=tAA sin(& CT + £7r), (22) 

approximately, by Art. 194 (15), and we may therefore replace (13) by 



*- 



<7* 



1 \ e kz cos (crt-km-lir) dk (23) 

r* J 



Comparing with (7) and (9) of Art. 241, and putting now z = 0, we find as 
the surface value of <f> 

» M ig7wg| " K " M (24) 

* Hobson, Proc. Lond. Math. Soc. xxv. 72, 73 (1893). This formula may, however, be dispensed 
with ; see the first footnote on p. 385 ante. 

t This result was given by Cauchy and Poisson. 



432 Surface Waves [chap, ix 

where k and a are to be expressed in terms of cr and t by means of (8) and 
(21). Note has here been taken of the fact that d 2 a/dk 2 is negative. Since 

at = (gkt 2 )$ = 2km, td 2 a/dk 2 = - ighk~% = - 2m z \gt 2 , . . .(25) 
we have <£o=-^ - — sin — (26) 

r 2%7TV 2 4*7 V ' 

The surface elevation is then given by (3). Keeping, for consistency, only the 
most important term, we find 

f_.jg_oo.tf, ( 27) 

2^7TOT 3 4ct 

which agrees with the result obtained, in other ways, by Cauchy and Poisson. 

It is not necessary to dwell on the interpretation, which will be readily 
understood from what has been said in Art. 240 with respect to the two- 
dimensional case. The consequences were worked out in some detail by 
Poisson on the hypothesis of an initial paraboloidal depression. 

When the initial data are of impulse, the typical solution is 

pcf> = cos at e kz J (km), \ 



.(28) 



?= sin at Jo (km), (" 

Hi J 

which, being generalized, gives, for the initial conditions 

P 4>o = F(m), f-0, (29) 



the solution 

d> = 

P 



I rco /-co 

6 = - cos ate 7 * J (km) kdk F (a) J (ka) a da, 
P J o Jo 

? = a sin at J (km) kdk \ F (a) J (ka)ada. 

gpJ o Jo ) 



...(30) 



gp 

In particular, for a concentrated impulse at the origin, such that 

\ C °F(a)27rada = l, (31) 

J o 

1 f 00 

we find £ = 0— cos at e kz Jo(km)kdk (32) 

lirp J o 

Since this may be written 

*-=i- If-— «*••/■.(*»)*# (33) 

Zirp ot J o 0" 

we find, performing 1/gp.d/dt on the results contained in (18) and (20), 

1 fPjQt) gt 2 2\P 2 („) (gt 2 ) 2 3\P 3 (ri 
9 ~~2iro\ r 2 2! r 8 4! r 4 



27rp [ r 
t 

2"rrpm 



- 8 > 5! W 9! \«J '"]' 



1- ...(34) 



255-256] Travelling Disturbance 433 

Again, when igt 2 /^ is large, we have, in place of (27), 

{>- -j£_Bn*£ (35)* 

256. We proceed to consider the effect of a local disturbance of pressure 
advancing with constant velocity over the surface f. This will give us, at all 
events as to the main features, an explanation of the peculiar system of waves 
which is seen to accompany a ship moving through sufficiently deep water. 

A complete investigation, after the manner of Arts. 242, 243, would be 
somewhat difficult; but the general characteristics can readily be made out 
with the help of preceding results, the procedure being similar to that of 
Art 249. 

Let us suppose that we have a pressure-point moving with velocity c 
along the axis of #, in the negative direction, and that at the instant under 
consideration it has reached the point 0. The elevation f at any point P may 





be regarded as due to a series of infinitely small impulses applied at equal 
infinitely short intervals at points of the axis of x to the right of 0. Of the 
annular wave-systems thus successively generated, those only will combine 
to produce a sensible effect at P which had their origin in the neighbourhood 
of certain points Q, which are determined by the consideration that the phase 
at P is 'stationary' for variations in the position of Q. Now if t is the time 
which the source of disturbance has taken to travel from Q to 0, the phase of 
the waves at P, originated at Q, is 

£+*-. w 

where ct = QP (Art. 255 (35)). Hence the condition for stationary phase is 

*=T < 2 > 

* The waves due to various types of explosive action beneath the surface have been studied 
by Terazawa, Proc. Roy. Soc. A, xcii. 57 (1915), and by the author of this work, I.e. ante p. 410, 
and Proc. Lond. Math. Soc. (2) xxi. 359 (1922). 

t For a more general treatment of such questions reference may be made to a paper by the 
author, "On Wave-Patterns due to a Travelling Disturbance," Phil. Mag. (6) xxxi. 539 (1916). 



434 



Surface Waves 



[chap. IX 



Since, in this differentiation, and P are regarded as fixed, we have 

is = c cos 6, 
where 6 = OQP ; hence 

0Q = ct = 2vsece (3) 

It is further evident that the points in the immediate neighbourhood 
of P, for which the resultant phase is the same as at P, will lie in a line 
perpendicular to QP. A glance at the figure on p. 433 then shews that a curve 
of uniform phase is characterized by the property that the tangent bisects 
the interval between the origin and the foot of the normal. If p denote the 
perpendicular from the origin to the tangent, and 6 the angle which p makes 
with the axis of %, we have, by a known formula, 



PZ=- 



whence 



dp 
dd ; 

p = a cos 2 6. 




.(6) 



The forms of the curves defined by (5) are shewn in the annexed figure*, 
which is traced from the equations 

dt) 

x=pcosd — ~sinO= J a (5 cos — cos 30), 
cLu 

y=psind + ~ cos 6 = — \a (sin V + sin SO), 
au 

* Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular 
Lectures, iii. 482], where a similar drawing is given. The investigation there referred to, based 
apparently on the theory of 'group-velocity,' was not published. See also E. E. Froude, "On 
Ship Eesistance," Papers of the Greenock Phil. Soc. Jan. 19, 1894. It is shewn immediately that 
there is a difference of phase between the two branches meeting at a cusp, so that the drawing 
does not represent quite accurately the configuration of the wave-ridges. 



256] 



Wave-Pattern 



435 



The phase-difference from one curve to the corresponding portion of the next 
is 2tt. This implies a difference 2-7rc 2 /<7 m the parameter a. 

Since two curves of the above kind pass through any assigned point P 
within the boundaries of the wave-system, it is evident that there are two 
corresponding effective positions of Q in the foregoing discussion. These are 
determined by a very simple construction. If the line OP be bisected in C, 
and a circle be drawn on GP as diameter, meeting the axis of a; in B, ly R 2 , 
the perpendiculars PQi, PQ2 to PRi, PR 2 > respectively, will meet the axis in 
the required points, Q lt Q 2 - For CR X is parallel to PQx and equal to %PQi', the 
perpendicular from on PR ± produced is therefore equal to PQ\. Similarly, 
the perpendicular from on PR 2 produced is equal to PQ 2 . 




The points Q ly Q 2 coincide when OP makes an angle sin -1 J , or 19° 28', with 
the axis of symmetry. For greater inclinations of OP they are imaginary. It 
appears also from (6) that the values of x, y are stationary when sin 2 = J ; 
this gives a series of cusps lying on the straight lines 

1 



^= + 



2V2 



= ± tan 19° 28' (7) 



To obtain an approximate estimate of the actual height of the waves, 
in the different parts of the system, we have recourse to the formula (35) of 
Art. 255. If P denote the total disturbing pressure, the elevation at P due 
to the annular wave-system started at a point Q to the right of may be 
written 

Sr=- o if.-4 -sin^.PoS<, (8) 



8V2^- Sm 4^ P »^ 



where 



*r=PQ, t = OQ/c. 

This is to be integrated with respect to t, but (as already explained) the only 
parts of the integral which contribute appreciably to the final result will be 
those for which t has very nearly the values (r 1} t 2 ) corresponding to the 
special points Q 1} Q 2 above mentioned. 

As regards the phase, we have, writing t = r + t' } 
9t\ 



+ t' 



dt VW J T 1 . 2 Idt 2 VW. 



+ . 



...(9) 



where, in the terms in [ ], t is to be put equal to tx or r 2 as the case may be. 



436 Surface Waves [chap, ix 

The second term vanishes by hypothesis, since the phase at P for waves 
started near Qi or Q 2 is ' stationary.' Again, we find 

!?l(9!l\- JL-9L • 9 t2 ( 2 * 2 *\ 

dt 2 \4m) ~ 2*r *r* V + 4 V *x 3 w *) ' 

c n c 2 sin 2 . 

Since <xr = ccos0, ot = , (10) 

'57 

this gives, with the help of (2), 

[1(g)] -£<*-*■*<> <"> 

Owing to the fluctuations of the trigonometrical term no great error 
will be committed if we neglect the variation of the first factor in (8), or if, 
further, we take the limits of integration with respect to t' to be ± oo . We 
have then, approximately, 

f=-8^/>(£>--<")* 

•00 f^n- 2 



J&/X&+-"')* < 12 > 



8 \J2TTplZ2 

where mj 2 = /- (i _ tan 2 X \ m 2 2 = ^- (tan 2 <9 2 - J), (13) 

ZtZTj Z'33'2 

and the suffixes refer to the points Q 1} Q 2 of the last figure. 

Since (°° cos m¥W==P° sin rnH'Ht' = */(&ir)lm, (14) 

J — 00 •/ —00 

where the positive value of m is understood, we find 

f_ ^fo .sin(^ +i7r ) gZggo ,sin(fA_ i7r ). 

8^2^*^^1*011 V4^i / 8V2ir*psr,*m s V 4ct 2 / 

(15) 

The two terms give the parts due to the transverse and lateral waves 
respectively. Since ^ x = PQ X = Jcri cos lf ct 2 = PQ2 — icr 2 cos 2 , it appears 
that if we consider either term by itself, the phase is constant along the 
corresponding part of the curve 

p = <oy = acos 2 0, 
whilst the elevation varies as 

V^iPo sec*fl 

7rV 3 a*' \/|l-3sin 2 <9| 

At the cusps, where the two systems combine, there is a phase-difference 
of a quarter-period between them. 

The formulae make f infinite at a cusp, where sin 2 #= J, but this is 
merely an indication of the failure of our approximation. That the elevation 
at a point P in the neighbourhood of a cusp would be relatively great might 
have been foreseen, since, as appears from (9) and (11), the range of points on 



256-256 a] Ship Waves . 437 

the axis of a; which have sent waves to P in sensibly the same phase is then 
abnormally extended. The infinity which occurs when 6 = \tt is of a some- 
what different character, being due to the artificial nature of the assumption 
we have made, of a pressure concentrated at a point. With a diffused pressure 
this difficulty would disappear*. 

It is to be noticed, moreover, that the whole of this investigation applies 
only to points for which gt 2 /^ is large ; cf. Arts. 240, 255. It will be found 
on examination that this restriction is equivalent to an assumption that the 
parameter a is large compared with 27rc 2 /g. The argument therefore does not 
apply without reserve to the parts of the wave-pattern near the origin. 

256 a. As already indicated, wave-systems of the above type are generated 
by other forms of travelling disturbance. Some of these cases are amenable to 
calculation. The translation of a submerged sphere, for instance, has been dealt 
with by Havelockf, and the wave-resistance determined. The writer J has 
discussed by another method the translation of a submerged solid, without 
restriction as to its precise form or orientation. The results are naturally 
simplest when the direction of motion coincides with one of the three directions 
of 'permanent translation' considered in Art. 124. The resistance is then 
given by the formula 

B _££±jWl (17) 

7T/3C 6 v y 

Here A denotes the appropriate inertia- coefficient from Art. 121, Q is the 
volume of the solid, c its velocity, and 

I=[*\ec 5 0e-W! c2 - seG * e dd y (18) 

Jo 

where / is the depth of immersion. Another form of this integral (due to 

Havelock) is 

/ = ie-"|&fo(«)+(l + ^) #i(<*)}, (19) 

in the accepted notation of Bessel Functions §, with a = gf/c 2 . For a sphere 
we have A = § 7rpa?, Q = f 7ra 3 , where a is the radius. Hence if M ' be the mass 
of fluid displaced, 

R = m'g.(fy 3 (yff.I, (20)|| 

in agreement with Havelock's result. As an example, if c = \/(gf), 

R = -365 M'g (a//) 8 . 

* More elaborate investigations have been carried out by Hopf in a dissertation of date Munich, 
1909, and Hogner, Arkiv for Matem. xvii. (1923). The latter writer examines in particular the 
shape of the waves near the 'cusps,' where the two systems cross. 

t Proc. Roy. Soc. A, xciii. 520 (1917); xcv. 354 (1918). See also Green, Phil. Mag. (6) xxxvi. 
48 (1918). 

% Proc. Boy. Soc. A, cxi. 14 (1926). 

§ Watson, p. 172. 

II This formula was given incorrectly in the author's paper. 



438 Surface Waves [chap, ix 

A graph of R as a function of c is given by Havelock; it has a general 
resemblance to the curve on p. 416. 

In a subsequent paper* the same method is applied by Havelock to a 
travelling disturbance consisting of various arrangements of (double) sources, 
with important applications to the wave-resistance of ships. 

Some further reference to the theoretical literature of wave-resistance may be in place 
here. Although the mode of disturbance is different, the action of the bows of a ship may 
be compared to that of a pressure-point. The diagram on p. 434 accounts for the two 
systems of transverse and lateral waves which are observed, and for the especially con- 
spicuous 'echelon' waves near the cusps, where the two systems cross. If in addition we 
imagine a negative pressure -point at the stern we get a rough representation of the action 
of the ship as a whole. With varying speeds the stern waves may tend partially to annul, 
or to reinforce, the effect of the bow waves, with the result that the resistance may be 
expected to fluctuate up and down as the length of the ship is increased, or the speed 
varied f. It is found in fact that the curve of resistance as a function of the speed exhibits 
several maxima (or 'humps') with the corresponding minima, as well as a general increase. 

To obtain an improved representation of what happens in the immediate neighbourhood 
of a ship and to calculate the consequent resistance is of course a difficult matter, but 
attempts have been made with considerable success. A beginning was made by J. H. 
Michelle with an idealized ship form, which differs mainly from that of a real ship in that 
the incliuation of the surface to the medial plane is everywhere small. This plan has 
recently been followed up by Wigley §, who has discussed a variety of forms (subject to the 
same limitation), calculated their resistance, and compared it with the results of model 
experiments, with a considerable measure of qualitative agreement. Havelock, in a long 
series of papers || has discussed the effect of various features in the design of a ship, such as 
length of 'parallel middle body,' mean draught, and so on. His method consists (in part) 
in the choice of a suitable arrangement of travelling sources, and is accordingly free from 
the special restriction above mentioned IT. 

A general formula for the wave-resistance of geometrically similar bodies, 
similarly immersed (wholly or partially), was given long ago by Froude. Since 
the resistance can only depend on the speed, the density of the fluid, the 
intensity of gravity, and on some linear magnitude which fixes the scale, 
considerations of dimensions shew that it must satisfy a relation of the form 

R=p^fi$). (2D 

where c is the speed, and I the characteristic linear magnitude. It will be 

* Proc. Roy. Soc. A, cxviii. 24 (1927). 

t W. Froude, "On the Effect on the Wave-Making Eesistance of Ships of Length of Parallel 
Middle Body," Trans. Inst. Nav. Arch. xvii. (1877). AlsoR. E. Froude, " On the Leading Phenomena 
of the Wave-Making Resistance of Ships," Trans. Inst. Nav. Arch. xxii. (1881), where drawings of 
actual wave-patterns under varied conditions of speed are given, which are, as to their main features, 
in striking agreement with the results of the above theory. Some of these drawings are reproduced 
in Kelvin's paper in the Proc. Inst. Mech. Eng. above cited. 

J Phil. Mag. (5) xlv. 106 (1898). 

§ Trans. Inst. Nav. Arch, lxviii. 124 (1926); lxix. 27 (1927); lxxii. (1930). 

|| 'In the Proc. Roy. Soc. from 1909 onwards. 

T[ Excellent accounts of the development of the subject are given by Hogner, Proc. Congress. 
App. Math. Delft, 1924, p. 146, and Wigley, Congress for techn. Mechanics, Stockolm, 1930. 



256 a-256 b] Effect of Limited Depth 439 

noticed that (17) is a particular case of this. It follows from (21) that the 
wave-resistance of a ship can be inferred from a model experiment provided 
the value of l/c 2 is the same on the model as on the full scale. 

256 b. To examine the modification produced in the wave-pattern when the 
depth of the water has to be taken into account, the argument on p. 433 
must be put in a more general form. If, as before, t is the time the pressure- 
point has taken to travel from Q to 0, it may be shewn that the phase of the 
disturbance at P, due to the impulse delivered at Q, will differ only by a 
constant from 

k(Vt-n), (22) 

where 27r/k is the predominant wave-length in the neighbourhood of P, 
and V the corresponding wave- velocity *. This predominant wave-length is 
determined by the condition that the phase is stationary for variations of the 
wave-length only, i.e. 

^r.k(Vt-vf) = O t or w=Ut, (23) 

where U, = d(kV)/dk, is the group- velocity (Art. 236). 

For the effective part of the disturbance at P, the phase (22) must 
further be stationary as regards variations in the position of Q; hence, 
differentiating partially with respect to t, we have 

bt = F, or F=ccos0, (24) 

since is = c cos 0. Now, referring to the figure on p. 433, we have 

p = ct cos 6 — nr = Vt — ts (25) 

Hence for a given wave-ridge p will bear a constant ratio to the wave- 
length X, and in passing from one wave-ridge to the next this ratio will 
increase (or decrease) by unity. Since \ is determined as a function of 6 by 
(24), this gives the relation between p and 0. 

Thus in the case of infinite depth, the formula (24) gives 

c 2 cos 2 <9 = F 2 = |^, (26) 

and the required relation is of the form 

p = acos 2 d, (27) 

as above. 

When the depth (h) is finite, we have 

c 2 cos 2 = F 2 = ^tanh^, (28) 

Lit A, 

and the relation is 

^tanh- = -%os 2 0, (29) 

a p gh x 

* The symbol c, which was previously employed in this sense, now denotes the velocity of 
the pressure-point over the water. 



440 Surface Waves [chap, ix 

where the values of a for successive wave-ridges are in arithmetic progression. 
Since the expression on the left-hand side cannot exceed unity, it appears 
that if c 2 > gh there will be an inferior limit to the value of 6, determined by 

cos 2 d = gh/c 2 , (30) 

the curve then extending to infinity. 

It follows that when the speed of the disturbing influence exceeds \/'{gh) 
the transverse waves disappear, and we have only the lateral waves. This 
tends to diminish the wave-making resistance (cf. Art. 249)*. 

The changes in the configuration of the wave-pattern as the ratio c 2 /gh 
increases from zero to infinity are traced by Havelockf. 

Standing Waves in Limited Masses of Water. 

257. The problem of free oscillations in two horizontal dimensions (%, y), 
in the case where the depth is uniform and the fluid is bounded laterally by 
vertical walls, can be reduced to the same analytical form as in Art. 190. 

If the origin be taken in the undisturbed surface, and if f denote the 
elevation at time t above this level, the conditions to be satisfied at the free 
surface are as in Art. 255 (3), (4). 

The equation of continuity, V 2 $ = 0, and the condition of zero vertical 
motion at the depth z — — h, are both satisfied by 

<p = fa cosh k(z + h), (1) 

where fa is a function of x, y, such that 

t&+t£+**-° (2 > 

The form of fa and the admissible values of k are determined by this equation, 
and by the condition that 

&-* (*> 

at the vertical walls. The corresponding values of the ' speed ' (<r) of the 
oscillations are then given by the surface-condition (6), of Art. 255 ; viz. we 
have 

a 2 = gh tanh Jch (4) 

This makes f = — sinhkh.fa (5) 

<T 

* It is found that the power required to propel a torpedo-boat in relatively shallow water 
increases with the speed up to a certain critical velocity, dependent on the depth, then decreases, 
and finally increases again. See papers by Easmussen, Trans. Inst. Nav. Arch. xli. 12 (1899); 
Bota, ibid. xlii. 239 (1900) ; Yarrow and Marriner, ibid, xlvii. 339, 344 (1905). 

f Proc. Roy. Soc. lxxxi. 426 (1908). See also Ekman, I.e. ante p. 371. 



256 b-257] Waves in Limited Masses of Water 441 

The conditions (2) and (3) are of the same form as in the case of small 
depth, and we could therefore at once write down the results for a rect- 
angular or a circular* tank. The values of k, and the forms of the free surface, 
in the various fundamental modes, are the same as in Arts. 190, 191 f, but 
the amplitude of the oscillation now diminishes with increasing depth below 
the surface, according to the law (1) ; whilst the speed of any particular mode 
is given by (4). 

When kh is small, we have <r 2 = k 2 gh, as in the Arts, referred to. 

We may also notice in this connection the case of a long and narrow rectangular tank 
having near its centre one or more cylindrical obstacles, whose generating lines are vertical. 



The origin being taken at the centre of the free surface, and the axis of x parallel to 
the length I, we imagine two planes x = ±x / to be drawn, such that x' is moderately large 
compared with the horizontal dimensions of the obstacles, whilst still small in comparison 
with the length (I). Beyond these planes we shall have 

3f+*k-a (e) 

approximately, and therefore, for x > x', 

(j) 1 = Asmkx + Bcoskx, ( 7 ) 

whilst, for x < - x', 

<f>i = A sin hx — B cos kx, (8) 

since in the gravest mode, which is alone here considered, <f> must be an odd function of x. 

In the region between the planes x=±x' the configuration of the lines <j> 1 = const, is, 
for a reason to be explained in Art. 290 in connection with other questions, sensibly the 
same as if in (2) we were to put £ = 0. So far as this region is concerned, the problem is 
in fact the same as that of conduction of electricity along a bar of metal which has the 
same form as the actual mass of water, and has accordingly one or more cylindrical 
perforations occupying the place of the obstacles. The electrical resistance between the 
two planes is then equivalent to that of a certain length 2x' + a of an unperforated bar of 
the same section. The difference of potential between the planes may be taken to be 
2(kAx' + B), by (7), since hod is small; and the current per unit sectional area is kA, 
approximately. Thus 

2(kAx' + B) = (2x' + a)kA, .... (9) 

whence B\A=\ha (10) 

and fa = A (sin hx + \ka cos kx\ (11) 

for x > x'. 

* For references to the original investigations by Poisson and Eayleigh of waves in a circular 
tank see p. 287. The problem was also treated by Merian, Ueber die Bewegung tropfbarer Fliis- 
sigkeiten in Gefassen, Basel, 1828 [see VonderMuhll, Math. Ann. xxvii. 575], and by Ostrogradsky, 
"Memoire sur la propagation des ondes dans un bassin cylindrique," Mem. des Sav. Etrang. 
iii. (1862). 

f It may be remarked that either of the two modes figured on p. 288 may easily be excited by 
properly timed horizontal agitation of a tumbler containing water. 



442 Surface Waves [chap, ix 

The condition d<f>/da;=0, to be satisfied for x=\l, gives 

cos^H — |^osinH, (12) 

or, since ka is a small quantity, 

cosp(Z+a)=0 (13) 

The introduction of the obstacles has therefore the effect of virtually increasing the length 
of the tank by a. The period of the gravest mode is accordingly 



=V(f coth T)> CM) 



where l'=l + a. 

The value of a is known for one or two cases. In the case of a circular column of radius 
6, in the centre of the tank, the formulae (11) and (13) of Art. 64 shew that fa varies as 
x+C, or x-\-rrb 2 /a, practically, when x is large compared with the breadth a of the tank. 
Comparing with (11) above we see that 

= 27r6>, (15) 

subject to the condition that the ratio bja must not exceed about £*. 

When the plane x—0 is occupied by a thin rigid diaphragm of breadth a, having a 
central vertical slit of breadth c> the formula is 

« = _logsec— ^-' (16) 

258. The number of cases of motion with a variable depth, of which the 
solution has been obtained, is very small. 

1°. We may notice, first, the two-dimensional oscillations of water across a channel 
whose section consists of two straight lines inclined at 45° to the vertical f. 

The axes of y, z being respectively horizontal and vertical, in the plane of a cross-section, 
we assume 

cj) + i\lr = A {cosh k (y + iz) + cos k (y+iz)}, (1) 

the time-factor cos(crt+e) being understood. This gives 

of) = A (cosh hy cos kz + cos ky cosh kz), \jr = A (sinh ky sin kz — sin hy sinh kz). . . .(2) 

The latter formula shews at once that the lines y=±z constitute the stream-line ^ = 0, 
and may therefore be taken as fixed boundaries. 

The condition to be satisfied at the free surface is, as in Art. 227, 

-**-*£ < 3 > 

Substituting from (2) we find, if h denote the height of the surface above the origin, 

a 2 (cosh ky cos kh + cos ky cosh kh) =gk ( — cosh ky sin kh + cos ky sinh kh). 

This will be satisfied for all values of y, provided 

<r 2 cosM= — gk sin kh, <r 2 cosh kh=gk sin kh, (4) 

whence tanhM= -tanM (5) 

This determines the admissible values of k ; the corresponding values of a are then given 
by either of the equations (4). 

* The formula (14) was in this case found to be in good agreement with experiment (Lamb 
and Cooke, Phil. Mag. (6) xx. 303 (1910)). The experiments were made chiefly with a view to 
test the above method of approximation, which has other more important applications; see 
Arts. 306, 307. 

f Kirchhoff, " Ueber stehende Schwingungen einer schweren Flussigkeit," Berl. Monatsber. 
May 15, 1879 [Ges. Abh. p. 428]; Greenhill, I.e. ante p. 372. 



257-258] Effect of Limited Depth 443 

Since (2) makes an even function of y, the oscillations which it represents are sym- 
metrical with respect to the medial plane y — 0. 

The asymmetrical oscillations are given by 

+ i\j, = iA {cosh k(y + iz)- cos k(y + iz)}, (6) 

or <f> = — A(smh.kysinkz+smkysinhkz), ty = A (cosh ky cos kz — cosky coshkz). ...(7) 

The stream-line \j/ = consists, as before, of the lines y = ±z; and the surface-condition (3) 
gives 

a 2 (sinh ky sin hh + sin hy sinh kh)=gk (sinh hy cos hh + sin ley cosh IcK). 

This requires 

cr 2 sin hh—gh cos M, o- 2 sinh kh—gh cosh M, (8) 

whence tanhM = tanM (9) 

The equations (5) and (9) present themselves in the theory of the lateral vibrations of 
a bar free at both ends ; viz. they are both included in the equation 

cos m cosh m = 1 , (10)* 

where m = 2M. 

The root M = 0, of (9), which is extraneous in the theory referred to, is now important; 
it corresponds in fact to the slowest mode of oscillation in the present problem. Putting 
Ak 2 =B, and making k infinitesimal, the formulae (7) become, on restoring the time-factor, 
and taking the real parts, 

(f>= -2Byz. cos (at + e\ ^r = B(y 2 -z 2 ). cos(o-* + e), (11) 

whilst from (8) o- 2 = | (12) 

The corresponding form of the free surface is 

f -£KL= 4 =^%- »*«+«) ( i3 > 

The surface in this mode is therefore always plane. The annexed figure shews the lines of 
motion (\j/ = const.) for a series of equidistant values of \jr. 




The next gravest mode is symmetrical, and is given by the lowest finite root of (5), 
which is M= 2-3650, whence or = 1*5244 (g/h)$. The profile of the surface has now two 

* Cf. Eayleigh, Ttieory of Sound, i. 277, where the numerical solution of the equation is 
fully discussed. 



444 Surface Waves [chap, ix 

nodes, whose positions are determined by putting <f> = 0, z = h, in (2); whence it is found 
that 

f=±-5516* 
The next mode corresponds to the lowest finite root of (9), and so ont. 

2°. Greenhill, in the paper already cited, has investigated the symmetrical oscillations 
of the water across a channel whose section consists of two straight lines inclined at 60° 
to the vertical. In the (analytically) simplest mode of this kind we have, omitting the 
time-factor. 

<t>+fy = iA(y + iz)* + B, (14) 

or <j>=Az(z 2 -3y*) + B, yJA = Ay(y 2 -'Sz 2 ), (15) 

the latter formula making \^=0 along the boundary y= +V3 . z. The surface-condition (3) 
is satisfied for z— h, provided 

<r 2 =g\h, B=2Ak* (16) 

The corresponding form of the free surface, viz. 



^Kl-*-T^-^-^+«). 



.(17) 



is a parabolic cylinder, with two nodes at distances of -5774 of the half-breadth from the 
centre. The slowest mode, which must evidently be of asymmetrical type, has not yet been 
determined. 

3°. If in any of the above cases we transfer the origin to either edge of the canal, and 
then make the breadth infinite, we get a system of standing waves on a sea bounded by a 
sloping bank. This may be regarded as made up of an incident and a reflected system. 
The reflection is complete, but there is in general a change of phase. 
When the inclination of the bank is 45° the solution is 

(j> = H {e he (cos ky - sin ky)+e~ k v (cos kz + sin kz)} cos (at + e) (18) 

For an inclination of 30° to the horizontal we have 

(f) = H{e k *sinky+e- hk(y/3y+z) sin$k(y-\/3z) 

-s/3e-* H « 3y - z) cos%k(y + sJSz)}cos(<rt + €) (19) 

In each case a 2 =gk, as in the case of waves on an unlimited sheet of deep water. 

These results, which may easily be verified ab initio, were given by KirchhofF (I.e.). 

259. An interesting problem which presents itself in this connection is 
that of the transversal oscillations of water contained in a canal of circular 
section. This has not yet been solved, but it may be worth while to point 
out that an approximate determination of the frequency of the slowest mode, 
in the case where the free surface is at the level of the axis, can be effected 
by Rayleigh's method, explained near the end of Art. 168. 

If we assume as an ' approximate type ' that in which the free surface remains always 
plane, making a small angle 6 (say) with the horizontal, it appears, from Art. 72 (17), that 
the kinetic energy T is given by 

2r -fi-i)^ 2 > m 

* Bayleigh, Theory of Sound, Art. 178. 

f An experimental verification of the frequencies, and of the positions of the loops (places of 
maximum vertical amplitude), in various fundamental modes, was made by Kirchhoff and 
Hansemann, "Ueber stehende Schwingungen des Wassers, " Wied. Ann. x. (1880) [Kirchhoff, 
Ges. Abh. p. 442]. 



258-260] Transverse Oscillations in a Channel 445 

where a is the radius, whilst for the potential energy V we have 

2V=$gpa36 2 (2) 

If we assume that 6 varies as cos (<rt + e\ this gives 



2_ 



8?r g 
48-3tt 2 a' 



.(3) 



whence o-=M69 (gla$*. 

In the case of a rectangular section of breadth 2a, and depth a, the speed is given by 
Art. 257 (4), where we must put k=ir/2a from Art. 178, and h=a. This gives 

(r 2 =^7rtanh^7r.^, (4) 

or o- = l'200 (g/a)*. The frequency in the actual problem is less, since the kinetic energy 
due to a given motion of the surface is greater, whilst the potential energy for a given 
deformation is the same. Cf. Art. 45. 

260. We may next consider the free oscillations of the water included 
between two transverse partitions in a uniform horizontal canal. Before 
proceeding to particular cases, we may examine for a moment the nature 
of the analytical problem. 

If the axis of x be parallel to the length, and the origin be taken in one 
of the ends, the velocity-potential in any one of the fundamental modes 
referred to may, by Fourier's Theorem, be supposed expressed in the form 

<f) — (P + Pi cos kx + P 2 cos 2kx + . . . + P s cos skx + ...) cos {at + e), . . .(1) 

where k = ir/l, if I denote the length of the compartment. The coefficients P s 
are here functions of y, z. If the axis of z be drawn vertically upwards, and 
that of y be therefore horizontal and transverse to the canal, the forms of 
these functions, and the admissible values of a, are to be determined from 
the equation of continuity 

VV = 0, (2) 

with the conditions that ^- ■ = (3) 

at the sides, and a 2 (j)=g ~ (4) 

at the free surface. Since d(f>/dx must vanish for x = and x = I, it follows 
from known principles^ that each term in (1) must satisfy the conditions (2), 
(3), (4) independently; viz. we must have 

^ + S-^ p -=°> ( 5 > 

with ^ = (6) 

on 

* Hydrodynamics, 2nd ed. (1895). Eayleigh finds, as a closer approximation, <r = l'1644 (g/a)^; 
see Phil. Mag. (5) xlvii. 566 (1899) [Papers, iv. 407]. 

t See Stokes, "On the Critical Values of the Sums of Periodic Series," Camb. Trans, viii. 
(1847) [Papers, i. 236] . 



446 Surface Waves [chap, ix 

at the lateral boundary, and 

**•-'!; < 7 > 

at the free surface. 

The terra P gives purely transverse oscillations such as have been dis- 
cussed in Art. 258. Any other term P g cos skx gives a series of fundamental 
modes with s nodal lines transverse to the canal, and 0, 1, 2, 3, ... nodal lines 
parallel to the length. 

It will be sufficient for our purpose to consider the term P x cos kx. It is 
evident that the assumption 

<f> = Pi cos kx . cos (at + e), (8) 

with a proper form of P 2 and the corresponding value of a determined as 
above, gives the velocity-potential of a possible system of standing waves, 
of arbitrary wave-length 27r/&, in an unlimited canal of the given form of 
section. Now, as explained in Art. 229, by superposition of two properly 
adjusted systems of standing waves of this type we can build up a system 
of progressive waves 

<j> = Pi cos (kx + at) (9) 

We infer that progressive waves of simple-harmonic profile, of any assigned 
wave-length, are possible in an infinitely long canal of any uniform section. 

We might go further, and assert the possibility of an infinite number of 
types, of any given wave-length, with wave-velocities ranging from a certain 
lowest value to infinity. The types, however, in which there are longitudinal 
nodes at a distance from the sides are from the present point of view of 
subordinate interest. 

Two extreme cases call for special notice, viz. where the wave-length is 
very great or very small compared with the dimensions of the transverse 
section. 

The most interesting types of the former class have no longitudinal nodes, 
and are covered by the general theory of 'long' waves given in Arts. 169, 170. 
The only additional information we can look for is as to the shapes of the 
wave-ridges in the direction transverse to the canal. 

In the case of relatively short waves, the most important type is one in 
which the ridges extend across the canal with gradually varying height, and 
the wave-velocity is that of free waves on deep water as given by Art. 229 (6). 

There is another type of short waves